Title: Spinning Tops
Author: Perry, John
Language: English
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THE EARL OF PEMBROKE TO THE ABBESS OF WILTON.

"Go spin, you jade! go spin!"

[Illustration: MAGNETISM, LIGHT, AND MOLECULAR SPINNING TOPS.

_Page 122._

_THE ROMANCE OF SCIENCE._

SPINNING TOPS.

_THE "OPERATIVES' LECTURE"_
OF THE BRITISH ASSOCIATION MEETING AT LEEDS,
6th SEPTEMBER, 1890.

BY

PROFESSOR JOHN PERRY,
M.E., D.Sc, LL.D., F.R.S.

With Numerous Illustrations.

_REPRINT OF NEW AND REVISED EDITION,_

_With an Illustrated Appendix on the Use of Gyrostats._

LONDON
SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE,
Northumberland Avenue, W.C.; 43, Queen Victoria Street, E.C.
BRIGHTON: 129, North Street.
NEW YORK: E. S. GORHAM.

1910

PUBLISHED UNDER THE DIRECTION OF THE GENERAL LITERATURE COMMITTEE

[_Date of last impression, April 1908_]

This Report of an Experimental Lecture
WAS INSCRIBED TO
THE LATE
LORD KELVIN,
BY HIS AFFECTIONATE PUPIL, THE LECTURER, WHO
HEREBY TOOK A CONVENIENT METHOD OF
ACKNOWLEDGING THE REAL AUTHOR OF
WHATEVER IS WORTH PUBLICATION
IN THE FOLLOWING
PAGES.

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PREFACE.

This is not the lecture as it was delivered. Instead of two pages of
letterpress and a woodcut, the reader may imagine that for half a minute
the lecturer played with a spinning top or gyrostat, and occasionally
ejaculated words of warning, admonition, and explanation towards his
audience. A verbatim report would make rather uninteresting reading, and I
have taken the liberty of trying, by greater fullness of explanation, to
make up to the reader for his not having seen the moving apparatus. It has
also been necessary in a treatise intended for general readers to simplify
the reasoning, the lecture having been delivered to persons whose life
experiences peculiarly fitted them for understanding scientific things. An
"argument" has been added at the end to make the steps of the reasoning
clearer.

  JOHN PERRY.

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{9}

SPINNING TOPS.

--------

At a Leeds Board School last week, the master said to his class, "There is
to be a meeting of the British Association in Leeds. What is it all about?
Who are the members of the British Association? What do they do?" There was
a long pause. At length it was broken by an intelligent shy boy: "Please,
sir, I know--they spin tops!"[1]

Now I am sorry to say that this answer was wrong. The members of the
British Association and the Operatives of Leeds have neglected top-spinning
since they were ten years of age. If more attention were paid to the
intelligent examination of the behaviour of tops, there would be greater
advances in mechanical engineering and a great many industries. There would
be a better general knowledge of astronomy. Geologists would not make
mistakes by millions of years, and our knowledge of Light, and Radiant
Heat, and other {10} Electro-magnetic Phenomena would extend much more
rapidly than it does.

I shall try to show you towards the end of the lecture that the fact of our
earth's being a spinning body is one which would make itself known to us
even if we lived in subterranean regions like the coming race of an
ingenious novelist.[2] It is the greatest and most persistent cause of many
of the phenomena which occur around us and beneath us, and it is probable
that even Terrestrial Magnetism is almost altogether due to it. Indeed
there is only one possible explanation of the _Vril-ya_ ignorance about the
earth's rotation. Their knowledge of mechanics and dynamics was immense; no
member attending the meeting of the British Association can approach them
in their knowledge of, I will not say, _Vril_, but even of quite vulgar
electricity and magnetism; and yet this great race which expresses so
strongly its contempt for Anglo-Saxon _Koom-Poshery_ was actually ignorant
of the fact that it had existed for untold generations inside an object
that spins about an axis.

Can we imagine for one instant that the children of that race had never
spun a top or trundled a hoop, and so had had no chance of being led to the
greatest study of nature? No; the only possible explanation lies in the
great novelist's never {11} having done these things himself. He had
probably as a child a contempt for the study of nature, he was a baby
Pelham, and as a man he was condemned to remain in ignorance even of the
powers of the new race that he had created.

The _Vril-ya_ ignorance of the behaviour of spinning bodies existing as it
does side by side with their deep knowledge of magnetism, becomes even more
remarkable when it comes home to us that the phenomena of magnetism and of
light are certainly closely connected with the behaviour of spinning
bodies, and indeed that a familiar knowledge of the behaviour of such
bodies is absolutely necessary for a proper comprehension of most of the
phenomena occurring in nature. The instinctive craving to investigate these
phenomena seems to manifest itself soon after we are able to talk, and who
knows how much of the intellectual inferiority of woman is due to her
neglect of the study of spinning tops; but alas, even for boys in the
pursuit of top-spinning, the youthful mind and muscle are left with no
other guidance than that which is supplied by the experience of young and
not very scientific companions. I remember distinctly that there were many
puzzling problems presented to me every day. There were tops which nobody
seemed able to spin, and there were others, well {12} prized objects, often
studied in their behaviour and coveted as supremely valuable, that behaved
well under the most unscientific treatment. And yet nobody, even the
makers, seemed to know why one behaved badly and the other well.

I do not disguise from myself the fact that it is rather a difficult task
to talk of spinning tops to men who have long lost that skill which they
wonder at in their children; that knowingness of touch and handling which
gave them once so much power over what I fear to call inanimate nature. A
problem which the child gives up as hopeless of solution, is seldom
attacked again in maturer years; he drives his desire for knowledge into
the obscure lumber-closets of his mind, and there it lies, with the
accumulating dust of his life, a neglected and almost forgotten instinct.
Some of you may think that this instinct only remains with those minds so
many of which are childish even to the limit of life's span; and probably
none of you have had the opportunity of seeing how the old dust rubs off
from the life of the ordinary man, and the old desire comes back to him to
understand the mysteries that surround him.

But I have not only felt this desire myself, I have seen it in the excited
eyes of the crowd of people who stand by the hour under the dropping
cherry-blossoms beside the red-pillared temple of {13} Asakusa in the
Eastern capital of Japan, watching the _tedzu-mashi_ directing the
evolutions of his heavily rimmed _Koma_. First he throws away from him his
great top obliquely into the air and catches it spinning on the end of a
stick, or the point of a sword, or any other convenient implement; he now
sends it about quite carelessly, catching it as it comes back to him from
all sorts of directions; he makes it run up the hand-rail of a staircase
into a house by the door and out again by the window; he makes it travel up
a great corkscrew. Now he seizes it in his hands, and with a few dexterous
twists gives it a new stock of spinning energy. He makes it travel along a
stretched string or the edge of a sword; he does all sorts of other curious
things with his tops, and suddenly sinks from his masterful position to beg
for a few coppers at the end of his performance.

How tame all this must seem to you who more than half forget your childish
initiation into the mysteries of nature; but trust me, if I could only make
that old top-spinner perform those magical operations of his on this
platform, the delight of the enjoyment of beautiful motion would come back.
Perhaps it is only in Japan that such an exhibition is possible; the land
where the waving bamboo, and the circling hawk, and the undulating summer
sea, and every beautiful motion of nature {14} are looked upon with
tenderness; and perhaps it is from Japan that we shall learn the
development of our childish enthusiasm.

The devotees of the new emotional art of beautiful motion and changing
colour are still in the main beggars like Homer, and they live in garrets
like Johnson and Savage; but the dawn of a new era is heralded, or rather
the dawn has already come, for Sir William Thomson's achievements in the
study of spinning tops rank already as by no means the meanest of his great
career.

If you will only think of it, the behaviour of the commonest spinning top
is very wonderful. When not spinning you see that it falls down at once, I
find it impossible to balance it on its peg; but what a very different
object it is when spinning; you see that it not only does not fall down, it
offers a strange resistance when I strike it, and actually lifts itself
more and more to an upright position. Once started on scientific
observation, nature gives us facts of an analogous kind in great plenty.

Those of you who have observed a rapidly moving heavy belt or rope, know
that rapid motion gives a peculiar quasi-rigidity to flexible and even to
fluid things.

Here, for example, is a disc of quite thin paper (Fig. 1), and when I set
it in rapid rotation you observe that it resists the force exerted by my
{15} hand, the blow of my fist, as if it were a disc of steel. Hear how it
resounds when I strike it with a stick. Where has its flexibility gone?

[Illustration: FIG. 1.]

Here again is a ring of chain which is quite flexible. It seems ridiculous
to imagine that this {16} could be made to stand up like a stiff hoop, and
yet you observe that when I give it a rapid rotation on this mandril and
let it slide off upon the table, it runs over the table just as if it were
a rigid ring, and when it drops on the floor it rebounds like a boy's hoop
(Fig. 2).

[Illustration: FIG. 2.]

Here again is a very soft hat, specially made for this sort of experiment.
You will note that it collapses to the table in a shapeless mass when I lay
it down, and seems quite incapable of resisting forces which tend to alter
its shape. In fact, there is almost a complete absence of rigidity; but
when this is spun on the end of a stick, first note {17} how it has taken a
very easily defined shape; secondly, note how it runs along the table as if
it were made of steel; thirdly, note how all at once it collapses again
into a shapeless heap of soft material when its rapid motion has ceased.
Even so you will see that when a drunken man is not leaning against a wall
or lamp-post, he feels that his only chance of escape from ignominious
collapse is to get up a decent rate of speed, to obtain a quasi-sobriety of
demeanour by rapidity of motion.

The water inside this glass vessel (Fig. 3) is in a state of rapid motion,
revolving with the vessel itself. Now observe the piece of paraffin wax A
immersed in the water, and you will see when I push at it with a rod that
it vibrates just as if it were surrounded with a thick jelly. Let us now
apply Prof. Fitzgerald's improvement on this experiment of Sir William
Thomson's. Here is a disc B stuck on the end of the rod; observe that when
I introduce it, although it does not touch A, A is repelled from the disc.
Now observe that when I twirl the disc it seems to attract A.

[Illustration: FIG. 3.[3]]

At the round hole in front of this box a rapid motion is given to a small
quantity of air which is mixed with smoke that you may see it. That
smoke-ring moves through the air almost like a solid body for a
considerable distance unchanged, and I am not sure that it may not be
possible yet {18} to send as a projectile a huge poisoned smoke-ring, so
that it may destroy or stupefy an army miles away. Remember that it is
really the same air all the time. You will observe that two smoke rings
sent from two boxes have curious actions {19} upon one another, and the
study of these actions has given rise to Thomson's smoke-ring or vortex
theory of the constitution of matter (Fig. 4).

[Illustration: FIG. 4.]

It was Rankine, the great guide of all engineers, who first suggested the
idea of molecular vortices in his explanations of heat phenomena and the
phenomena of elasticity--the idea that every particle of matter is like a
little spinning top; but I am now speaking of Thomson's theory. To imagine
that an atom of matter is merely a {20} curiously shaped smoke-ring formed
miraculously in a perfect fluid, and which can never undergo permanent
alteration, looks to be a very curious and far-fetched hypothesis. But in
spite of certain difficulties, it is the foundation of the theory which
will best explain most of the molecular phenomena observed by philosophers.
Whatever be the value of the theory, you see from these experiments that
motion does give to small quantities of fluid curious properties of
elasticity, attraction and repulsion; that each of these entities refuses
to be cut in two; that you cannot bring a knife even near the smoke-ring;
and that what may be called a collision between two of them is not very
different in any way from the collision between two rings of india-rubber.

Another example of the rigidity given to a fluid by rapid motion, is the
feeling of utter helplessness which even the strongest swimmers sometimes
experience when they get caught in an eddy underneath the water.

I could, if I liked, multiply these instances of the quasi-rigidity which
mere motion gives to flexible or fluid bodies. In Nevada a jet of water
like the jet from a fireman's hose, except that it is much more rapid,
which is nearly as easily projected in different directions, is used in
mining, and huge masses of earth and rock are rapidly disintegrated {21} by
the running water, which seems to be rather like a bar of steel than a jet
of water in its rigidity.

It is, however, probable that you will take more interest in this box of
brass which I hold in my hands. You see nothing moving, but really, inside
this case there is a fly-wheel revolving rapidly. Observe that I rest this
case on the table on its sharp edge, a sort of skate, and it does not
tumble down as an ordinary box would do, or as this box will do after a
while, when its contents come to rest. Observe that I can strike it violent
blows, and it does not seem to budge from its vertical position; it turns
itself just a little round, but does not get tilted, however hard I strike
it. Observe that if I do get it tilted a little it does not fall down, but
slowly turns with what is called a precessional motion (Fig. 5).

You will, I hope, allow me, all through this lecture, to use the term
_precessional_ for any motion of this kind. Probably you will object more
strongly to the great liberty I shall take presently, of saying that the
case _precesses_ when it has this kind of motion; but I really have almost
no option in the matter, as I must use some verb, and I have no time to
invent a less barbarous one.

[Illustration: FIG. 5.]

When I hold this box in my hands (Fig. 6), I find that if I move it with a
motion of mere translation in any direction, it feels just as it would do
{22} if its contents were at rest, but if I try to turn it in my hands I
find the most curious great resistance to such a motion. The result is that
when you hold this in your hands, its readiness to move so long as it is
not turned round, and its great resistance to turning round, and its
unexpected tendency to turn in a different way from that in which you try
to turn it, give one the most uncanny sensations. It seems almost as if an
invisible being had hold of the box and exercised forces capriciously. And
{23} indeed there is a spiritual being inside, what the algebraic people
call an impossible quantity, what other mathematicians call "an operator."

[Illustration: FIG. 6.]

Nearly all the experiments, even the tops and other apparatus you have seen
or will see to-night, have been arranged and made by my enthusiastic
assistant, Mr. Shepherd. The following experiment is not only his in
arrangement; even the idea of it is his. He said, you may grin and contort
your body with that large gyrostat in your hands, but many of your audience
will simply say to {24} themselves that you only _pretend_ to find a
difficulty in turning the gyrostat. So he arranged this pivoted table for
me to stand upon, and you will observe that when I now try to turn the
gyrostat, it will not turn; however I may exert myself, it keeps pointing
to that particular corner of the room, and all my efforts only result in
turning round my own body and the table, but not the gyrostat.

Now you will find that in every case this box only resists having the axis
of revolution of its hidden flywheel turned round, and if you are
interested in the matter and make a few observations, you will soon see
that every spinning body like the fly-wheel inside this case resists more
or less the change of direction of its spinning axis. When the fly-wheels
of steam-engines and dynamo machines and other quick speed machines are
rotating on board ship, you may be quite sure that they offer a greater
resistance to the pitching or rolling or turning of the ship, or any other
motion which tends to turn their axes in direction, than when they are not
rotating.

Here is a top lying on a plate, and I throw it up into the air; you will
observe that its motion is very difficult to follow, and nobody could
predict, before it falls, exactly how it will alight on the plate; it may
come down peg-end foremost, or hindmost, or sideways. But when I spin it
(Fig. 7), and now throw it up into the air, there is no doubt whatever {25}
as to how it will come down. The spinning axis keeps parallel to itself,
and I can throw the top up time after time, without disturbing much the
spinning motion.

[Illustration: FIG. 7.]

[Illustration: FIG. 8.]

If I pitch up this biscuit, you will observe that I can have no certainty
as to how it will come down, but if I give it a spin before it leaves my
hand there is no doubt whatever (Fig. 8). Here is a hat. I throw it up, and
I cannot be sure as to how it will move, but if I give it a spin, you see
that, as {26} with the top and the biscuit, the axis about which the
spinning takes place keeps parallel to itself, and we have perfect
certainty as to the hat's alighting on the ground brim downwards (Fig. 9).

[Illustration: FIG. 9.]

I need not again bring before you the very soft hat to which we gave a
quasi-rigidity a few minutes ago; but you will remember that my assistant
sent that off like a projectile through the air when it was spinning, and
that it kept its spinning axis parallel to itself just like this more rigid
hat and the biscuit.

[Illustration: FIG. 10.]

[Illustration: FIG. 11.]

I once showed some experiments on spinning tops to a coffee-drinking,
tobacco-smoking audience in that most excellent institution, the Victoria
Music Hall in London. In that music hall, things are not very different
from what they are at any other {27} music hall except in beer, wine, and
spirits being unobtainable, and in short scientific addresses being
occasionally given. Now, I impressed my audience as strongly as I could
with the above fact, that if one wants to throw a quoit with certainty as
to how it will alight, one gives it a spin; if one wants to throw a hoop or
a hat to somebody to catch upon a stick, one gives the hoop or hat a spin;
the disinclination of a spinning body to let its axis get altered in
direction can always be depended upon. I told them that this was why
smooth-bore guns cannot be depended upon for accuracy;[4] that the spin
which an ordinary bullet took depended greatly on how it chanced to touch
the muzzle as it just left the gun, whereas barrels are now rifled, that
is, spiral grooves are now cut inside the barrel of a gun, and excrescences
from the bullet or projectile fit into these grooves, so that as it is
forced along the barrel of the gun by the explosive force of the powder, it
must also spin about its axis. Hence it leaves the gun with a perfectly
well-known spinning motion about which there can be no doubt, and we know
too that Fig. 10 shows the {28} kind of motion which it has afterwards,
for, just like the hat or the biscuit, its spinning axis keeps nearly
parallel to itself. Well, this was all I could do, for I am not skilful in
throwing hats or quoits. But after my address was finished, and after a
young lady in a spangled dress had sung a comic song, two jugglers came
upon the stage, and I could not have had better illustrations of the above
principle than were given in almost every trick performed by this lady and
gentleman. They sent hats, and hoops, and plates, and umbrellas spinning
from one to the other. One of them threw a stream of knives into the air,
catching them and throwing them up again with perfect precision and my now
educated audience shouted with delight, and showed in other unmistakable
{29} ways that they observed the spin which that juggler gave to every
knife as it left his hand, so that he might have a perfect knowledge as to
how it would come back to him again (Fig. 11). {30} It struck me with
astonishment at the time that, almost without exception, every juggling
trick performed that evening was an illustration of the above principle.
And now, if you doubt my statement, just ask a child whether its hoop is
more likely to tumble down when it is rapidly rolling along, or when it is
going very slowly; ask a man on a bicycle to go more and more slowly to see
if he keeps his balance better; ask a ballet-dancer how long she could
stand on one toe without balancing herself with her arms or a pole, if she
were not spinning; ask astronomers how many months would elapse before the
earth would point ever so far away from the pole star if it were not
spinning; and above all, ask a boy whether his top is as likely to stand
upright upon its peg when it is not spinning as when it is spinning.

[Illustration: FIG. 12.]

We will now examine more carefully the behaviour of this common top (Fig.
12). It is not {31} spinning, and you observe that it tumbles down at once;
it is quite unstable if I leave it resting upright on its peg. But now note
that when it is spinning, it not only will remain upright resting on its
peg, but if I give it a blow and so disturb its state, it goes circling
round with a precessional motion which grows gradually less and less as
time goes on, and the top lifts itself to the upright position again. I
hope you do not think that time spent in careful observation of a
phenomenon of this kind is wasted. Educated observation of the commonest
phenomena occurring in our everyday life is never wasted, and I often feel
that if workmen, who are the persons most familiar with inorganic nature,
could only observe and apply simple scientific laws to their observations,
instead of a great discovery every century we should have a great discovery
every year. Well, to return to our top; there are two very curious
observations to make. Please neglect for a short time the slight wobbling
motions that occur. One observation we make is, that the top does not at
first bow down in the direction of the blow. If I strike towards the south,
the top bows towards the west; if I strike towards the west, the top bows
down towards the north. Now the reason of this is known to all scientific
men, and the principle underlying the top's behaviour is of very great {32}
importance in many ways, and I hope to make it clear to you. The second
fact, that the top gradually reaches its upright position again, is one
known to everybody, but the reason for it is not by any means well known,
although I think that you will have no great difficulty in understanding
it.

The first phenomenon will be observed in this case which I have already
shown you. This case (Fig. 5), {33} with the fly-wheel inside it, is called
a _gyrostat_. When I push the case it does not bow down, but slowly turns
round. This gyrostat will not exhibit the second phenomenon; it will not
rise up again if I manage to get it out of its upright position, but, on
the contrary, will go precessing in wider and wider circles, getting
further and further away from its upright position.

[Illustration: FIG. 13.]

[Illustration: FIG. 14.]

The first phenomenon is most easily studied in this balanced gyrostat (Fig.
13). You here see the fly-wheel G in a strong brass frame F, which is
supported so that it is free to move about the vertical axis A B, or about
the horizontal axis C D. The gyrostat is balanced by a weight W. Observe
that I can increase the leverage of W or diminish it by shifting the
position of the sleeve at A so that it will tend to either lift or lower
the gyrostat, or exactly balance it as it does now. You must observe
exactly what it is that we wish to study. If I endeavour to push F
downwards, with the end of this stick (Fig. 14), it really moves
horizontally to the right; now I push it to the right (Fig. 15), and it
only rises; now push it up, and you see that it goes to the left; push it
to the left, and it only goes downwards. You will notice that if I clamp
the instrument so that it cannot move vertically, it moves at once
horizontally; if I prevent mere horizontal motion it readily moves
vertically when I push it. Leaving it free as {34} before, I will now shift
the position of the weight W, so that it tends continually to lift the
gyrostat, and of course the instrument does not lift, it moves horizontally
with a slow precessional motion. I now again shift the weight W, so that
the gyrostat would fall if it were not spinning (Fig. 16), and it now moves
horizontally with a slow precessional motion which is in a direction
opposed to the last. These phenomena are easily explained, but, {35} as I
said before, it is necessary first to observe them carefully. You all know
now, vaguely, the fundamental fact. It is that if I try to make a very
quickly spinning body change the direction of its axis, the direction of
the axis will change, but not in the way I intended. It is even more
curious than my countryman's pig, for when he wanted the pig to go to Cork,
he had to pretend that he was driving the pig home. His rule was a very
{36} simple one, and we must find a rule for our spinning body, which is
rather like a crab, that will only go along the road when you push it
sidewise.

[Illustration: FIG. 15.]

[Illustration: FIG. 16.[5]]

[Illustration: FIG. 10.]

As an illustration of this, consider the spinning projectile of Fig. 10.
The spin tends to keep its axis always in the same direction. But there is
a defect in the arrangement, which you are now in a {37} position to
understand. You see that at A the air must be pressing upon the
undersurface A A, and I have to explain that this pressure tends to make
the projectile turn itself broadside on to the air. A boat in a current not
allowed to move as a whole, but tied at its middle, sets itself broadside
on to the current. Observe this disc of cardboard which I drop through the
air edgewise, and note how quickly it sets itself broadside on and falls
more slowly; and some of you may have thrown over into the water at Aden
small pieces of silver for the diving boys, and you are aware that if it
were not for this slow falling of the coins with a wobbling motion
broadside on, it would be nearly impossible for any diving boy to get
possession of them. Now all this is a parenthesis. The {38} pressure of the
air tends to make the projectile turn broadside on, but as the projectile
is spinning it does not tilt up, no more than this gyrostat does when I try
to tilt it up, it really tilts out of the plane of the diagram, out of the
plane of its flight; and only that artillerymen know exactly what it will
do, this kind of _windage_ of the projectile would give them great trouble.

You will notice that an experienced child when it wants to change the
direction of a hoop, just exerts a tilting pressure with its hoop-stick. A
man on a bicycle changes his direction by leaning over so as to be out of
balance. It is well to remind you, however, that the motion of a bicycle
and its rider is not all rotational, so that it is not altogether the
analogue of a top or gyrostat. The explanation of the swerving from a
straight path when the rider tilts his body, ultimately comes to the same
simple principle, Newton's second law of motion, but it is arrived at more
readily. It is for the same reason--put briefly, the exercise of a
centripetal force--that when one is riding he can materially assist his
horse to turn a corner quickly, if he does not mind appearances, by
inclining his body towards the side to which he wants to turn; and the more
slowly the horse is going the greater is the tendency to turn for a given
amount of tilting of one's body. Circus-riders, when galloping in a circle,
assist their horses greatly by the position of their bodies; it is {39} not
to save themselves from falling by centrifugal force that they take a
position on a horse's back which no riding-master would allow his pupil to
imitate; and the respectable riders of this country would not scorn to help
their horses in this way to quick turning movements, if they had to chase
and collect cattle like American cowboys.

Very good illustrations of change of direction are obtained in playing
_bowls_. You know that a bowl, if it had no _bias_, that is, if it had no
little weight inside it tending to tilt it, would roll along the level
bowling-green in a straight path, its speed getting less and less till it
stopped. As a matter of fact, however, you know that at the beginning, when
it is moving fast, its path is pretty straight, but because it always has
bias the path is never quite straight, and it bends more and more rapidly
as the speed diminishes. In all our examples the slower the spin the
quicker is the precession produced by given tilting forces.

Now close observation will give you a simple rule about the behaviour of a
gyrostat. As a matter of fact, all that has been incomprehensible or
curious disappears at once, if instead of speaking of this gyrostat as
moving up or down, or to the right or left, I speak of its motions about
its various axes. It offers no resistance to mere motion of translation.
But when I spoke of its moving {40} horizontally, I ought to have said that
it moved about the vertical axis A B (Fig. 13). Again, what I referred to
as up and down motion of F is really motion in a vertical plane about the
horizontal axis C D. In future, when I speak of trying to give motion to F,
think only of the axis about which I try to turn it, and then a little
observation will clear the ground.

[Illustration: FIG. 17.]

[Illustration: FIG. 18.]

Here is a gyrostat (Fig. 17), suspended in gymbals so carefully that
neither gravity nor any frictional forces at the pivots constrain it;
nothing that I can do to this frame which I hold in my hand will affect the
direction of the axis E F of the gyrostat. Observe that I whirl round on my
toes like a ballet-dancer while this is in my hand. I move it about in all
sorts of ways, but if it was pointing to the pole star at the beginning it
remains pointing to the pole star; if it pointed towards the moon at the
beginning it still points {41} towards the moon. The fact is, that as there
is almost no frictional constraint at the pivots there are almost no forces
tending to turn the axis of rotation of the gyrostat, and I can only give
it motions of translation. But now I will clamp this vertical spindle by
means of a screw and repeat my ballet-dance whirl; you will note that I
need not whirl round, a very small portion of a whirl is enough to cause
this gyrostat (Fig. 18) to set its spinning axis vertical, to set its axis
parallel to the vertical axis of rotation which I give it. Now I whirl in
the opposite direction, the gyrostat at once turns a somersault, turns
completely round and remains again with its axis vertical, and if you were
to carefully note the direction of the spinning of the {42} gyrostat, you
would find the following rule to be generally true:--Pay no attention to
mere translational motion, think only of rotation about axes, and just
remember that when you constrain the axis of a spinning body to rotate, it
will endeavour to set its own axis parallel to the new axis about which you
rotate it; and not only is this the case, but it will endeavour to have the
direction of its own spin the same as the direction of the new rotation. I
again twirl on my toes, holding this frame, and now I know that to a person
looking down upon the gyrostat and me from the ceiling, as I revolved in
the direction of the hands of a clock, the gyrostat is spinning in the
direction of the hands of a clock; but if I revolve against the clock
direction (Fig. 19) the gyrostat tumbles over so as again to be revolving
in the same direction as that in which I revolve.

[Illustration: FIG. 19.]

This then is the simple rule which will enable you to tell beforehand how a
gyrostat will move {43} when you try to turn it in any particular
direction. You have only to remember that if you continued your effort long
enough, the spinning axis would become parallel to your new axis of motion,
and the direction of spinning would be the same as the direction of your
new turning motion.

Now let me apply my rule to this balanced gyrostat. I shove it, or give it
an impulse downwards, but observe that this really means a rotation about
the horizontal axis C D (Fig. 13), and hence the gyrostat turns its axis as
if it wanted to become parallel to C D. Thus, looking down from above (as
shown by Fig. 20), O E was the direction of the spinning axis, O D was the
axis about which I endeavoured to move it, and the instantaneous effect was
that O E altered to the position O G. A greater impulse of the same kind
would have caused the spinning axis instantly to go to O H or O J, whereas
an upward opposite impulse would have instantly made the spinning axis
point in the direction O K, O L or O M, depending on how great the impulse
was and the rate of spinning. When one observes these phenomena for the
first time, one says, "I shoved it down, and it moved to the right; I
shoved it up, and it moved to the left;" but if the direction of the spin
were opposite to what it is, one would say, "I shoved it down, and it moved
to the left; I shoved it up, and it moved to the right." The simple {44}
statement in all cases ought to be, "I wanted to rotate it about a new
axis, and the effect was to send its spinning axis towards the direction of
the new axis." And now if you play with this balanced gyrostat as I am
doing, shoving it about in all sorts of ways, you will find the rule to be
a correct one, and there is no difficulty in predicting what will happen.

[Illustration: FIG. 20.]

{45}

If this rule is right, we see at once why precession takes place. I put
this gyrostat (Fig. 13) out of balance, and if it were not rotating it
would fall downwards; but a force acting downwards really causes the
gyrostat to move to the right, and so you see that it is continually moving
in this way, for the force is always acting downwards, and the spinning
axis is continually chasing the new axes about which gravity tends
continually to make it revolve. We see also why it is that if the want of
balance is the other way, if gravity tends to lift the gyrostat, the
precession is in the opposite direction. And in playing with this gyrostat
as I do now, giving it all sorts of pushes, one makes other observations
and sees that the above rule simplifies them all; that is, it enables us to
remember them. For example, if I use this stick to hurry on the precession,
the gyrostat moves in opposition to the force which causes the precession.
I am particularly anxious that you should remember this. At present the
balance-weight is so placed that the gyrostat would fall if it were not
spinning. But it is spinning, and so it precesses. If gravity were greater
it would precess faster, and it comes home to us that it is this precession
which enables the force of gravity to be inoperative in mere downward
motion. You see that if the precession is hurried, it is more than
sufficient to balance gravity, {46} and the gyrostat rises. If I retard the
precession, it is unable to balance gravity, and the gyrostat falls. If I
clamp this vertical axis so that precession is impossible, you will notice
that the gyrostat falls just as if it were not spinning. If I clamp the
instrument so that it cannot move vertically, you notice how readily I can
make it move horizontally; I can set it rotating horizontally like any
ordinary body.

In applying our rule to this top, observe that the axis of spinning is the
axis E F of the top (Fig. 12). As seen in the figure, gravity is tending to
make the top rotate about the axis F D, and the spinning axis in its chase
of the axis F D describes a cone in space as it precesses. This gyrostat,
which is top-heavy, rotates and precesses in much the same way as the top;
that is, if you apply our rule, or use your observation, you will find that
to an observer above the table the spinning and precession occur in the
same direction, that is, either both with the hands of a watch, or both
against the hands of a watch. Whereas, a top like this before you (Fig.
21), supported above its centre of gravity, or the gyrostat here (Fig. 22),
which is also supported above its centre of gravity, or the gyrostat shown
in Fig. 56, or any other gyrostat supported in such a way that it would be
in stable equilibrium if it were not spinning; in all these {47} cases, to
an observer placed above the table, the precession is in a direction
opposite to that of the spinning.

[Illustration: FIG. 21.]

[Illustration: FIG. 22.]

{48}

If an impulse be given to a top or gyrostat in the direction of the
precession, it will rise in opposition to the force of gravity, and should
at any instant the precessional velocity be greater than what it ought to
be for the balance of the force of gravity, the top or gyrostat will rise,
its precessional velocity diminishing. If the precessional velocity is too
small, the top will fall, and as it falls the precessional velocity
increases.

Now I say that all these facts, which are mere facts of observation, agree
with our rule. I wish I dare ask you to remember them all. You will observe
that in this wall sheet I have made a list of them. I speak of gravity as
causing the precession, but the forces may be any others than such as are
due to gravity.

WALL SHEET.

I. RULE. When forces act upon a spinning body, tending to cause rotation
about any other axis than the spinning axis, the spinning axis sets itself
in better agreement with the new axis of rotation. Perfect agreement would
mean perfect parallelism, the directions of rotation being the same.

II. Hurry on the precession, and the body rises in opposition to gravity.
{49}

III. Delay the precession and the body falls, as gravity would make it do
if it were not spinning.

IV. A common top precesses in the same direction as that in which it spins.

V. A top supported above its centre of gravity, or a body which would be in
stable equilibrium if not spinning, precesses in the opposite direction to
that of its spinning.

VI. The last two statements come to this:--When the forces acting on a
spinning body tend to make the _angle_ of precession greater, the
precession is in the same direction as the spinning, and _vice versâ_.

Having by observation obtained a rule, every natural philosopher tries to
make his rule a rational one; tries to explain it. I hope you know what we
mean when we say that we explain a phenomenon; we really mean that we show
the phenomenon to be consistent with other better known phenomena. Thus
when you unmask a spiritualist and show that the phenomena exhibited by him
are due to mere sleight-of-hand and trickery, you explain the phenomena.
When you show that they are all consistent with well-observed and
established mesmeric influences, you are also said to explain the
phenomena. When you show that they can be effected by means of telegraphic
messages, or by reflection of light from mirrors, you explain the {50}
phenomena, although in all these cases you do not really know the nature of
mesmerism, electricity, light, or moral obliquity.

The meanest kind of criticism is that of the man who cheapens a scientific
explanation by saying that the very simplest facts of nature are
unexplainable. Such a man prefers the chaotic and indiscriminate wonder of
the savage to the reverence of a Sir Isaac Newton.

[Illustration: FIG. 23.]

The explanation of our rule is easy. Here is a gyrostat (Fig. 23) something
like the earth in shape, and it is at rest. I am sorry to say that I am
compelled to support this globe in a very visible manner by gymbal rings.
If this globe were just floating in the air, if it had no tendency to fall,
my explanation would be easier to understand, and I could illustrate it
better experimentally. Observe the point P. If I move the globe slightly
about the axis A, the point P moves to Q. But suppose instead of this that
the globe and inner gymbal {51} ring had been moved about the axis B; the
point P would have moved to R. Well, suppose both those rotations took
place simultaneously. You all know that the point P would move neither to Q
nor to R, but it would move to S; P S being the diagonal of the little
parallelogram. The resultant motion then is neither about the axis O A in
space, nor about the axis O B, but it is about some such axis as O C.

To this globe I have given two rotations simultaneously. Suppose a little
being to exist on this globe which could not see the gymbals, but was able
to observe other objects in the room. It would say that the direction of
rotation is neither about O A nor about O B, but that the real axis of its
earth is some line intermediate, O C in fact.

If then a ball is suddenly struck in two different directions at the same
instant, to understand how it will spin we must first find how much spin
each blow would produce if it acted alone, and about what axis. A spin of
three turns per second about the axis O A (Fig. 24), and a spin of two
turns per second about the axis O B, really mean that the ball will spin
about the axis O C with a spin of three and a half turns per second. To
arrive at this result, I made O A, 3 feet long (any other scale of
representation would have been right) {52} and O B, 2 feet long, and I
found the diagonal O C of the parallelogram shown on the figure to be 3½
feet long.

Observe that if the rotation about the axis O A is _with_ the hands of a
watch looking from O to A, the rotation about the axis O B looking from O
to B, must also be with the hands of a watch, and the resultant rotation
about the axis O C is also in a direction with the hands of a watch looking
from O to C. Fig. 25 shows in two diagrams how necessary it is that on
looking from O along either O A or O B, the rotation should be in the same
direction as regards the hands of a watch. These constructions are well
known to all who have studied elementary mechanical principles. Obviously
if the rotation about O A is very much greater than the rotation about O B,
then the position of the new axis O C must be much nearer O A than O B.

[Illustration: FIG. 24.]

[Illustration: FIG. 25.]

We see then that if a body is spinning about an axis O A, and we apply
forces to it which {53} would, if it were at rest, turn it about the axis O
B; the effect is to cause the spinning axis to be altered to O C; that is,
the spinning axis sets itself in better agreement with the new axis of
rotation. This is the first statement on our wall sheet, the rule from
which all our other statements are derived, assuming that they were not
really derived from observation. Now I do not say that I have here given a
complete proof for all cases, for the fly-wheels in these gyrostats are
running in bearings, and the bearings constrain the axes to take the new
positions, whereas there is no such {54} constraint in this top; but in the
limited time of a popular lecture like this it is not possible, even if it
were desirable, to give an exhaustive proof of such a universal rule as
ours is. That I have not exhausted all that might be said on this subject
will be evident from what follows.

If we have a spinning ball and we give to it a new kind of rotation, what
will happen? Suppose, for example, that the earth were a homogeneous
sphere, and that there were suddenly impressed upon it a new rotatory
motion tending to send Africa southwards; the axis of this new spin would
have its pole at Java, and this spin combined with the old one would cause
the earth to have its true pole somewhere between the present pole and
Java. It would no longer rotate about its present axis. In fact the axis of
rotation would be altered, and there would be no tendency for anything
further to occur, because a homogeneous sphere will as readily rotate about
one axis as another. But if such a thing were to happen to this earth of
ours, which is not a sphere but a flattened spheroid like an orange, its
polar diameter being the one-third of one per cent. shorter than the
equatorial diameter; then as soon as the new axis was established, the axis
of symmetry would resent the change and would try to become again the axis
of rotation, and a great wobbling motion would ensue. {55} I put the matter
in popular language when I speak of the resentment of an axis; perhaps it
is better to explain more exactly what I mean. I am going to use the
expression Centrifugal Force. Now there are captious critics who object to
this term, but all engineers use it, and I like to use it, and our captious
critics submit to all sorts of ignominious involution of language in
evading the use of it. It means the force with which any body acts upon its
constraints when it is constrained to move in a curved path. The force is
always directed away from the centre of the curve. When a ball is whirled
round in a curve at the end of a string its centrifugal force tends to
break the string. When any body keyed to a shaft is revolving with the
shaft, it may be that the centrifugal forces of all the parts just balance
one another; but sometimes they do not, and then we say that the shaft is
out of balance. Here, for example, is a disc of wood rotating. It is in
balance. But I stop its motion and fix this piece of lead, A, to it, and
you observe when it rotates that it is so much out of balance that the
bearings of the shaft and the frame that holds them, and even the
lecture-table, are shaking. Now I will put things in balance again by
placing another piece of lead, B, on the side of the spindle remote from A,
and when I again rotate the disc (Fig. 26) there {56} is no longer any
shaking of the framework. When the crank-shaft of a locomotive has not been
put in balance by means of weights suitably placed on the driving-wheels,
there is nobody in the train who does not feel the effects. Yes, and the
coal-bill shows the effects, for an unbalanced engine tugs the train
spasmodically instead of exerting an efficient steady pull. My friend
Professor Milne, of Japan, places earthquake measuring instruments on
engines and in trains for measuring this and other wants of balance, and he
has shown unmistakably that two engines of nearly the same general design,
one balanced properly and the other not, consume very different amounts of
coal in making the same journey at the same speed.

[Illustration: FIG. 26.]

If a rotating body is in balance, not only does the axis of rotation pass
through the centre of gravity (or rather centre of mass) of the body, but
{57} the axis of rotation must be one of the three principal axes through
the centre of mass of the body. Here, for example, is an ellipsoid of wood;
A A, B B, and C C (Fig. 27) are its three principal axes, and it would be
in balance if it rotated about any one of these three axes, and it would
not be in balance if it rotated about any other axis, unless, indeed, it
were like a homogeneous sphere, every diameter of which is a principal
axis.

[Illustration: FIG. 27.]

Every body has three such principal axes through its centre of mass, and
this body (Fig. 27) has them; but I have here constrained it to rotate
about the axis D D, and you all observe the effect of the unbalanced
centrifugal forces, which is nearly great enough to tear the framework in
pieces. The higher the speed the more important this want of balance is. If
the speed is doubled, the centrifugal forces become four times as great;
and modern mechanical engineers with their quick speed engines, some of
which revolve, like the fan-engines of torpedo-boats, at 1700 revolutions
per minute, require to pay great attention to this subject, which the older
engineers never troubled their {58} heads about. You must remember that
even when want of balance does not actually fracture the framework of an
engine, it will shake everything, so that nuts and keys and other
fastenings are pretty sure to get loose.

I have seen, on a badly-balanced machine, a securely-fastened pair of nuts,
one supposed to be locking the other, quietly revolving on their bolt at
the same time, and gently lifting themselves at a regular but fairly rapid
rate, until they both tumbled from the end of the bolt into my hand. If my
hand had not been there, the bolts would have tumbled into a receptacle in
which they would have produced interesting but most destructive phenomena.
You would have somebody else lecturing to you to-night if that event had
come off.

Suppose, then, that our earth were spinning about any other axis than its
present axis, the axis of figure. If spun about any diameter of the equator
for example, centrifugal forces would just keep things in a state of
unstable equilibrium, and no great change might be produced until some
accidental cause effected a slight alteration in the spinning axis, and
after that the earth would wobble very greatly. How long and how violently
it would wobble, would depend on a number of circumstances about which I
will not now venture to guess. If you {59} tell me that on the whole, in
spite of the violence of the wobbling, it would not get shaken into a new
form altogether, then I know that in consequence of tidal and other
friction it would eventually come to a quiet state of spinning about its
present axis.

You see, then, that although every body has three axes about which it will
rotate in a balanced fashion without any tendency to wobble, this balance
of the centrifugal forces is really an unstable balance in two out of the
three cases, and there is only one axis about which a perfectly stable
balanced kind of rotation will take place, and a spinning body generally
comes to rotate about this axis in the long run if left to itself, and if
there is friction to still the wobbling.

To illustrate this, I have here a method of spinning bodies which enables
them to choose as their spinning axis that one principal axis about which
their rotation is most stable. The various bodies can be hung at the end of
this string, and I cause the pulley from which the string hangs to rotate.
Observe that at first the disc (Fig. 28 _a_) rotates soberly about the axis
A A, but you note the small beginning of the wobble; now it gets quite
violent, and now the disc is stably and smoothly rotating about the axis B
B, which is the most important of its principal axes. {60}

[Illustration: FIG. 28.]

Again, this cone (Fig. 28 _b_) rotates smoothly at first about the axis A
A, but the wobble begins and gets very great, and eventually the cone
rotates smoothly about the axis B B, which is the most important of its
principal axes. Here again is a rod hung from one end (Fig. 28 _d_).

See also this anchor ring. But you may be more interested in this limp ring
of chain (Fig. 28 _c_). See how at first it hangs from the cord vertically,
and how the wobbles and vibrations end in its becoming a perfectly circular
ring lying all in a horizontal plane. This experiment illustrates also the
quasi-rigidity given to a flexible body by rapid motion.

To return to this balanced gyrostat of ours (Fig. 13). It is not
precessing, so you know that the weight W just balances the gyrostat F. Now
if I leave the instrument to itself after I give a downward impulse to F,
not exerting merely a steady pressure, you will notice that F swings to the
right for the reason already given; but it swings too fast and too far,
just like any other swinging body, and it is easy from what I have already
said, to see that this wobbling motion (Fig. 29) should be the result, and
that it should continue until friction stills it, and F takes its permanent
new position only after some time elapses.

You see that I can impose this wobble or nodding {62} motion upon the
gyrostat whether it has a motion of precession or not. It is now nodding as
it processes round and round--that is, it is rising and falling as it
precesses.

[Illustration: FIG. 29.]

Perhaps I had better put the matter a little more clearly. You see the same
phenomenon in this top. If the top is precessing too fast for the force of
gravity the top rises, and the precession diminishes in consequence; the
precession being now too slow to balance gravity, the top falls a little
and the {63} precession increases again, and this sort of vibration about a
mean position goes on just as the vibration of a pendulum goes on till
friction destroys it, and the top precesses more regularly in the mean
position. This nodding is more evident in the nearly horizontal balanced
gyrostat than in a top, because in a top the turning effect of gravity is
less in the higher positions.

When scientific men try to popularize their discoveries, for the sake of
making some fact very plain they will often tell slight untruths, making
statements which become rather misleading when their students reach the
higher levels. Thus astronomers tell the public that the earth goes round
the sun in an elliptic path, whereas the attractions of the planets cause
the path to be only approximately elliptic; and electricians tell the
public that electric energy is conveyed through wires, whereas it is really
conveyed by all other space than that occupied by the wires. In this
lecture I have to some small extent taken advantage of you in this way; for
example, at first you will remember, I neglected the nodding or wobbling
produced when an impulse is given to a top or gyrostat, and, all through, I
neglect the fact that the instantaneous axis of rotation is only nearly
coincident with the axis of figure of a precessing gyrostat or top. And
indeed you may generally {64} take it that if all one's statements were
absolutely accurate, it would be necessary to use hundreds of technical
terms and involved sentences with explanatory, police-like parentheses; and
to listen to many such statements would be absolutely impossible, even for
a scientific man. You would hardly expect, however, that so great a
scientific man as the late Professor Rankine, when he was seized with the
poetic fervour, would err even more than the popular lecturer in making his
accuracy of statement subservient to the exigencies of the rhyme as well as
to the necessity for simplicity of statement. He in his poem, _The
Mathematician in Love_, has the following lines--

 "The lady loved dancing;--he therefore applied
    To the polka and waltz, an equation;
  But when to rotate on his axis he tried,
  His centre of gravity swayed to one side,
    And he fell by the earth's gravitation."

Now I have no doubt that this is as good "dropping into poetry" as can be
expected in a scientific man, and ----'s science is as good as can be
expected in a man who calls himself a poet; but in both cases we have
illustrations of the incompatibility of science and rhyming.

[Illustration: FIG. 17.]

The motion of this gyrostat can be made even more complicated than it was
when we had {65} nutation and precession, but there is really nothing in it
which is not readily explainable by the simple principles I have put before
you. Look, for example, at this well-balanced gyrostat (Fig. 17). When I
strike this inner gymbal ring in any way you see that it wriggles quickly
just as if it were a lump of jelly, its rapid vibrations dying away just
like the rapid vibrations of any yielding elastic body. This strange
elasticity is of very great interest when we consider it in relation to the
molecular properties of matter. Here again (Fig. 30) we have an example
which is even more interesting. I have supported the cased {66} gyrostat of
Figs. 5 and 6 upon a pair of stilts, and you will observe that it is moving
about a perfectly stable position with a very curious staggering kind of
vibratory motion; but there is nothing in these motions, however curious,
that you cannot easily explain if you have followed me so far.

[Illustration: FIG. 30.]

Some of you who are more observant than the others, will have remarked that
all these precessing gyrostats gradually fall lower and lower, just as they
would do, only more quickly, if they were not spinning. And if you cast
your eye upon the third statement of our wall sheet (p. 49) you will
readily understand why it is so.

"Delay the precession and the body falls, as gravity would make it do if it
were not spinning." {67} Well, the precession of every one of these is
resisted by friction, and so they fall lower and lower.

I wonder if any of you have followed me so well as to know already why a
spinning top rises. Perhaps you have not yet had time to think it out, but
I have accentuated several times the particular fact which explains this
phenomenon. Friction makes the gyrostats fall, what is it that causes a top
to rise? Rapid rising to the upright position is the invariable sign of
rapid rotation in a top, and I recollect that when quite vertical we used
to say, "She sleeps!" Such was the endearing way in which the youthful
experimenter thought of the beautiful object of his tender regard.

All so well known as this rising tendency of a top has been ever since tops
were first spun, I question if any person in this hall knows the
explanation, and I question its being known to more than a few persons
anywhere. Any great mathematician will tell you that the explanation is
surely to be found published in _Routh_, or that at all events he knows men
at Cambridge who surely know it, and he thinks that he himself must have
known it, although he has now forgotten those elaborate mathematical
demonstrations which he once exercised his mind upon. I believe that all
such statements are made in error, but I cannot {68} be sure.[6] A partial
theory of the phenomenon was given by Mr. Archibald Smith in the _Cambridge
Mathematical Journal_ many years ago, but the problem was solved by Sir
William Thomson and Professor Blackburn when they stayed together one year
at the seaside, reading for the great Cambridge mathematical examination.
It must have alarmed a person interested in Thomson's success to notice
that the seaside holiday was really spent by him and his friend in spinning
all sorts of rounded stones which they picked up on the beach.

And I will now show you the curious phenomenon that puzzled him that year.
This ellipsoid (Fig. 31) will represent a waterworn stone. It is lying in
its most stable state on the table, and I give it a spin. You see that for
a second or two it was inclined to go on spinning about the axis A A, but
it began to wobble violently, and after a while, when these wobbles
stilled, you saw that it was spinning nicely with its axis B B vertical;
but then a new series of wobblings began and became more violent, and when
they ceased you saw that the object had at length reached a settled state
of {69} spinning, standing upright upon its longest axis. This is an
extraordinary phenomenon to any person who knows about the great
inclination of this body to spin in the very way in which I first started
it spinning. You will find that nearly any rounded stone when spun will get
up in this way upon its longest axis, if the spin is only vigorous enough,
and in the very same way this spinning top tends to get more and more
upright.

[Illustration: FIG. 31.]

I believe that there are very few mathematical explanations of phenomena
which may not be given in quite ordinary language to people who have an
ordinary amount of experience. In most cases the symbolical algebraic
explanation must be given first by somebody, and then comes the time for
its translation into ordinary language. This is the foundation of the new
thing called Technical Education, which assumes that a {70} workman may be
taught the principles underlying the operations which go on in his trade,
if we base our explanations on the experience which the man has acquired
already, without tiring him with a four years' course of study in
elementary things such as is most suitable for inexperienced children and
youths at public schools and the universities.

[Illustration: FIG. 32.]

[Illustration: FIG. 33.]

With your present experience the explanation of the rising of the top
becomes ridiculously simple. If you look at statement _two_ on this wall
sheet (p. 48) and reflect a little, some of you will be able, without any
elaborate mathematics, to give the simple reason for this that Thomson gave
me sixteen years ago. "Hurry on the precession, and the body rises in
opposition to gravity." Well, as I am not touching the top, and as the body
does rise, we look at once for something that is hurrying on the
precession, and we naturally look to the way in which its peg is rubbing on
the table, for, with the exception of the atmosphere this top is touching
nothing else than the table. Observe carefully how any of these objects
precesses. Fig. 32 shows the way in which a top spins. Looked at from
above, if the top is spinning in the direction of the hands of a watch, we
know from the fourth statement of our wall sheet, or by mere observation,
that it also precesses in the direction of the hands {71} of a watch; that
is, its precession is such as to make the peg roll at B into the paper. For
you will observe that the peg is rolling round a circular path on the
table, G being nearly motionless, and the axis A G A describing nearly a
cone in space whose vertex is G, above the table. Fig. 33 {72} shows the
peg enlarged, and it is evident that the point B touching the table is
really like the bottom of a wheel B B', and as this wheel is rotating, the
rotation causes it to roll _into_ the paper, away from us. But observe that
its mere precession is making it roll _into_ the paper, and that the spin
if great enough wants to roll the top faster than the precession lets it
roll, so that it hurries on the precession, and therefore the top rises.
That is the simple explanation; the spin, so long as it is {73} great
enough, is always hurrying on the precession, and if you will cast your
recollection back to the days of your youth, when a top was supported on
your hand as this is now on mine (Fig. 34), and the spin had grown to be
quite small, and was unable to keep the top upright, you will remember that
you dexterously helped the precession by giving your hand a circling motion
so as to get from your top the advantages as to uprightness of a slightly
longer spin.

[Illustration: FIG. 34.]

I must ask you now by observation, and the application of exactly the same
argument, to explain the struggle for uprightness on its longer axis of any
rounded stone when it spins on a table. I may tell you that some of these
large rounded-looking objects which I now spin before you in illustration,
are made hollow, and they are either of wood or zinc, because I have not
the skill necessary to spin large solid objects, and yet I wanted to have
objects which you would be able to see. This small one (Fig. 31) is the
largest solid one to which my fingers are able to give sufficient spin.
Here is a very interesting object (Fig. 35), spherical {74} in shape, but
its centre of gravity is not exactly at its centre of figure, so when I lay
it on the table it always gets to its position of stable equilibrium, the
white spot touching the table as at A. Some of you know that if this sphere
is thrown into the air it seems to have very curious motions, because one
is so apt to forget that it is the motion of its centre of gravity which
follows a simple path, and the boundary is eccentric to the centre of
gravity. Its motions when set to roll upon a carpet are also extremely
curious.

[Illustration: FIG. 35.]

Now for the very reasons that I have already given, when this sphere is
made to spin on the table, it always endeavours to get its white spot
uppermost, as in C, Fig. 35; to get into the position in which when not
spinning it would be unstable.

[Illustration: FIG. 36.]

The precession of a top or gyrostat leads us at once to think of the
precession of the great spinning body on which we live. You know that the
earth {75} spins on its axis a little more than once every twenty-four
hours, as this orange is revolving, and that it goes round the sun once in
a year, as this orange is now going round a model sun, or as is shown in
the diagram (Fig. 36). Its spinning axis points in the direction shown,
very nearly to the star which is called the pole star, almost infinitely
far away. In the figure and model I have greatly exaggerated the elliptic
nature of the earth's path, as is quite usual, although it may be a little
misleading, because the earth's path is much more nearly circular than many
people imagine. As a matter of fact the earth is about three million miles
nearer the sun in winter than it is in summer. This seems at first
paradoxical, but we get to understand it when we reflect that, because of
the slope of the earth's axis to the ecliptic, we people who live in the
northern hemisphere have the sun less vertically above us, and have a
shorter day in the winter, and hence each square foot of our part of the
earth's surface receives much less heat every day, and so we feel colder.
Now in about 13,000 years the earth will have precessed just half a
revolution (_see_ Fig. 38); the axis will then be sloped towards the sun
when it is nearest, instead of away from it as it is now; consequently we
shall be much warmer in summer and colder in winter than we are now. Indeed
we shall then be much worse off than the southern {77} hemisphere people
are now, for they have plenty of oceanic water to temper their climate. It
is easy to see the nature of the change from figures 36, 37, and 38, or
from the model as I carry the orange and its symbolic knitting-needle round
the model sun. Let us imagine an observer placed above this model, far
above the north pole of the earth. He sees the earth rotating against the
direction of the hands of a watch, and he finds that it precesses with the
hands of a watch, so that spin and precession are in opposite directions.
Indeed it is because of this that we have the word "precession," which we
now apply to the motion of a top, although the precession of a top is in
the same direction as that of the spin.

[Illustration: FIG. 37.]

[Illustration: FIG. 38.]

The practical astronomer, in explaining the _luni-solar precession of the
equinoxes_ to you, will not probably refer to tops or gyrostats. He will
tell you that the _longitude_ and _right ascension_ of a star seem to
alter; in fact that the point on the ecliptic from which he makes his
measurements, namely, the spring equinox, is slowly travelling round the
ecliptic in a direction opposite to that of the earth in its orbit, or to
the apparent path of the sun. The spring equinox is to him for heavenly
measurements what the longitude of Greenwich is to the navigator. He will
tell you that aberration of light, and parallax of the stars, {80} but more
than both, this precession of the equinoxes, are the three most important
things which prevent us from seeing in an observatory by transit
observations of the stars, that the earth is revolving with perfect
uniformity. But his way of describing the precession must not disguise for
you the physical fact that his phenomenon and ours are identical, and that
to us who are acquainted with spinning tops, the slow conical motion of a
spinning axis is more readily understood than those details of his
measurements in which an astronomer's mind is bound up, and which so often
condemn a man of great intellectual power to the life of drudgery which we
generally associate with the idea of the pound-a-week cheap clerk.

[Illustration: FIG. 22.]

The precession of the earth is then of the same nature as that of a
gyrostat suspended above its centre of gravity, of a body which would be
stable and not top-heavy if it were not spinning. In fact the precession of
the earth is of the same nature as that of this large gyrostat (Fig. 22),
which is suspended in gymbals, so that it has a vibration like a pendulum
when not spinning. I will now spin it, so that looked at from above it goes
against the hands of a watch, and you observe that it precesses with the
hands of a watch. Here again is a hemispherical wooden ship, in which there
is a gyrostat with its axis vertical. It is in stable {81} equilibrium.
When the gyrostat is not spinning, the ship vibrates slowly when put out of
equilibrium; when the gyrostat is spinning the ship gets a motion of
precession which is opposite in direction to that of the spinning.
Astronomers, beginning with Hipparchus, have made observations of the
earth's motion for us, and we have observed the motions of gyrostats, and
we naturally seek for an explanation of the precessional motion of the
earth. The equator of the earth makes an angle of 23½° with the ecliptic,
which is the plane of the earth's orbit. Or the spinning axis of the earth
is always at angle of 23½° with a perpendicular to the ecliptic, and makes
a complete revolution in 26,000 years. The surface of the water on which
this wooden ship is floating represents the ecliptic. The axis {82} of
spinning of the gyrostat is about 23½° to the vertical; the precession is
in two minutes instead of 26,000 years; and only that this ship does not
revolve in a great circular path, we should have in its precession a pretty
exact illustration of the earth's precession.

The precessional motion of the ship, or of the gyrostat (Fig. 22), is
explainable, and in the same way the earth's precession is at once
explained if we find that there are forces from external bodies tending to
put its spinning axis at right angles to the ecliptic. The earth is a
nearly spherical body. If it were exactly spherical and homogeneous, the
resultant force of attraction upon it, of a distant body, would be in a
line through its centre. And again, if it were spherical and
non-homogeneous, but if its mass were arranged in uniformly dense,
spherical layers, like the coats of an onion. But the earth is not
spherical, and to find what is the nature of the attraction of a distant
body, it has been necessary to make pendulum observations all over the
earth. You know that if a pendulum does not alter in length as we take it
about to various places, its time of vibration at each place enables the
force of gravity at each place to be determined; and Mr. Green proved that
if we know the force of gravity at all places on the surface of the earth,
although we may know nothing about the {83} state of the inside of the
earth, we can calculate with absolute accuracy the force exerted by the
earth on matter placed anywhere outside the earth; for instance, at any
part of the moon's orbit, or at the sun. And hence we know the equal and
opposite force with which such matter will act on the earth. Now pendulum
observations have been made at a great many places on the earth, and we
know, although of course not with absolute accuracy, the attraction on the
earth, of matter outside the earth. For instance, we know that the
resultant attraction of the sun on the earth is a force which does not pass
through the centre of the earth's mass. You may comprehend the result
better if I refer to this diagram of the earth at midwinter (Fig. 39), and
use a popular method of description. A and B may roughly be called the
protuberant parts of the earth--that protuberant belt of matter which makes
the {84} earth orange-shaped instead of spherical. On the spherical portion
inside, assumed roughly to be homogeneous, the resultant attraction is a
force through the centre.

[Illustration: FIG. 39.]

I will now consider the attraction on the protuberant equatorial belt
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 23½°.
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 23½° 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 23½° with the earth's equator, and the
moon's orbit makes an angle 5½° with the ecliptic, we see that the moon's
orbit sometimes makes an angle of 29° with the earth's equator, and
sometimes only 18°, changing from 29° to 18°, and back to 29° 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 5½° 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
23½°, 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 5½° 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.

Now let us consider what occurs. The room is rotating about the earth's
axis, and we know the rate of rotation; but we only want to know for our
present purpose how much of the total rotation is about a vertical line in
the room. If the room were at the North Pole, the whole rotation would be
about the vertical line. If the room were at the equator, none of its
rotation would be about a vertical line. In our latitude now, the
horizontal rate of rotation about a vertical axis is about four-fifths of
the whole rate of rotation of the earth on its axis, and this is the amount
that would be measured by our microscope. This experiment would give no
result at a place on the equator, but in our latitude you would have a
laboratory proof of the rotation of the earth. Foucault made the
measurements with great accuracy.

If you now clamp the frame, and allow the spinning axis to have no motion
except in a horizontal plane, the motion which the earth tends to give it
about a vertical axis cannot now affect the gyrostat, but the earth
constrains it to move about an axis due north and south, and consequently
the spinning axis tries to put itself parallel {108} to the north and south
direction (Fig. 47). Hence with such an instrument it is easy to find the
true north. If there were absolutely no friction the instrument would
vibrate about the true north position like the compass needle (Fig. 50),
although with an exceedingly slow swing.

[Illustration: FIG. 47.]

It is with a curious mixture of feelings that one first recognizes the fact
that all rotating bodies, fly-wheels of steam-engines and the like, are
always tending to turn themselves towards the pole star; gently and vainly
tugging at their foundations {109} to get round towards the object of their
adoration all the time they are in motion.

[Illustration: FIG. 48.]

Now we have found the meridian as in Fig. 47, we can begin a third
experiment. Prevent motion horizontally, that is, about a vertical axis,
but give the instrument freedom to move vertically in the meridian, like a
transit instrument in an observatory {110} about its horizontal axis. Its
revolution with the earth will tend to make it change its angular position,
and therefore it places itself parallel to the earth's axis; when in this
position the daily rotation no longer causes any change in its direction in
space, so it continues to point to the pole star (Fig. 48). It would be an
interesting experiment to measure with a delicate chemical balance the
force with which the axis raises itself, and in this way _weigh_ the
rotational motion of the earth.[9]

Now let us turn the frame of the instrument G B round a right angle, so
that the spinning axis can only move in a plane at right angles to the
meridian; obviously it is constrained by the vertical component of the
earth's rotation, and points vertically downwards.

[Illustration: FIG. 49.]

[Illustration: FIG. 50.]

This last as well as the other phenomena of which I have spoken is very
suggestive. Here is a magnetic needle (Fig. 49), sometimes called a dipping
needle from the way in which it is suspended. If I turn its {111} frame so
that it can only move at right angles to the meridian, you see that it
points vertically. You may reflect upon the analogous properties of this
magnetic needle (Fig. 50) and of the gyrostat (Fig. 47); they both, when
only capable of moving horizontally, point to the north; and you see that a
very frictionless gyrostat might be used as a compass, or at all events as
a corrector of compasses.[10] I have just put before you another analogy,
and I want you to understand that, although these are only analogies, they
are not mere chance analogies, for there is undoubtedly a dynamical
connection between the magnetic and the gyrostatic phenomena. Magnetism
depends on rotatory motion. The molecules of matter are in actual rotation,
and a certain allineation of the axes of the rotations produces what we
call magnetism. In a steel bar not magnetized the little axes of rotation
are all in different directions. The process {112} of magnetization is
simply bringing these rotations to be more or less round parallel axes, an
allineation of the axes. A honey-combed mass with a spinning gyrostat in
every cell, with all the spinning axes parallel, and the spins in the same
direction, would--I was about to say, would be a magnet, but it would not
be a magnet in all its properties, and yet it would resemble a magnet in
many ways.[11]

[Illustration: FIG. 51.]

[Illustration: FIG. 52.]

Some of you, seeing electromotors and other electric contrivances near this
table, may think that they have to do with our theories and explanations of
magnetic phenomena. But I must explain that this electromotor which I hold
in my hand (Fig. 51) is used by me merely as the {113} most convenient
means I could find for the spinning of my tops and gyrostats. On the
spindle of the motor is fastened a circular piece of wood; by touching this
key I can supply the motor with electric energy, and the wooden disc is now
rotating very rapidly. I have only to bring its rim in contact with any of
these tops or gyrostats to set them spinning, and you see that I can set
half a dozen gyrostats a-spinning in a few seconds; this chain of
gyrostats, for instance. Again, this larger motor (Fig. 52), too large to
move about in my hand, is fastened to the table, and I have used {114} it
to drive my larger contrivances; but you understand that I use these just
as a barber might use them to brush your hair, or Sarah Jane to clean the
knives, or just as I would use a little steam-engine if it were more
convenient for my purpose. It was more convenient for me to bring from
London this battery of accumulators and these motors than to bring sacks of
coals, and boilers, and steam-engines. But, indeed, all this has the deeper
meaning that we can give to it if we like. Love is as old as the hills, and
every day Love's messages are carried by the latest servant of man, the
telegraph. These spinning tops were known probably to primeval man, and yet
we have not learnt from them more than the most fractional portion of the
lesson that they are always sending out to an unobservant world. Toys like
these were spun probably by the builders of the Pyramids when they were
boys, and here you see them side by side with the very latest of man's
contrivances. I feel almost as Mr. Stanley might feel if, with the help of
the electric light and a magic-lantern, he described his experiences in
that dreadful African forest to the usual company of a London drawing-room.

The phenomena I have been describing to you play such a very important part
in nature, that if time admitted I might go on expounding and {115}
explaining without finding any great reason to stop at one place rather
than another. The time at my disposal allows me to refer to only one other
matter, namely, the connection between light and magnetism and the
behaviour of spinning tops.

You are all aware that sound takes time to travel. This is a matter of
common observation, as one can see a distant woodchopper lift his axe again
before one hears the sound of his last stroke. A destructive sea wave is
produced on the coast of Japan many hours after an earthquake occurs off
the coast of America, the wave motion having taken time to travel across
the Pacific. But although light travels more quickly than sound or wave
motion in the sea, it does not travel with infinite rapidity, and the
appearance of the eclipse of one of Jupiter's satellites is delayed by an
observable number of minutes because light takes time to travel. The
velocity has been measured by means of such observations, and we know that
light travels at the rate of about 187,000 miles per second, or thirty
thousand millions of centimetres per second. There is no doubt about this
figure being nearly correct, for the velocity of light has been measured in
the laboratory by a perfectly independent method.

Now the most interesting physical work done since Newton's time is the
outcome of the experiments of Faraday and the theoretical deductions of
{116} Thomson and Maxwell. It is the theory that light and radiant heat are
simply electro-magnetic disturbances propagated through space. I dare not
do more than just refer to this matter, although it is of enormous
importance. I can only say, that of all the observed facts in the sciences
of light, electricity, and magnetism, we know of none that is in opposition
to Maxwell's theory, and we know of many that support it. The greatest and
earliest support that it had was this. If the theory is correct, then a
certain electro-magnetic measurement ought to result in exactly the same
quantity as the velocity of light. Now I want you to understand that the
electric measurement is one of quantities that seem to have nothing
whatever to do with light, except that one uses one's eyes in making the
measurement; it requires the use of a two-foot rule and a magnetic needle,
and coils of wire and currents of electricity. It seemed to bear a
relationship to the velocity of light, which was not very unlike the fabled
connection between Tenterden Steeple and the Goodwin Sands. It is a
measurement which it is very difficult to make accurately. A number of
skilful experimenters, working independently, and using quite different
methods, arrived at results only one of which is as much as five per cent.
different from the observed velocity of light, and some of them, {117} on
which the best dependence may be placed, agree exactly with the average
value of the measurements of the velocity of light.

There is then a wonderful agreement of the two measurements, but without
more explanation than I can give you now, you cannot perhaps understand the
importance of this agreement between two seemingly unconnected magnitudes.
At all events we now know, from the work of Professor Hertz in the last two
years, that Maxwell's theory is correct, and that light is an
electro-magnetic disturbance; and what is more, we know that
electro-magnetic disturbances, incomparably slower than red-light or heat,
are passing now through our bodies; that this now recognized kind of
radiation may be reflected and refracted, and yet will pass through brick
and stone walls and foggy atmospheres where light cannot pass, and that
possibly all military and marine and lighthouse signalling may be conducted
in the future through the agency of this new and wonderful kind of
radiation, of which what we call light is merely one form. Why at this
moment, for all I know, two citizens of Leeds may be signalling to each
other in this way through half a mile of houses, including this hall in
which we are present.[12]

{118}

I mention this, the greatest modern philosophical discovery, because the
germ of it, which was published by Thomson in 1856, makes direct reference
to the analogy between the behaviour of our spinning-tops and magnetic and
electrical phenomena. It will be easier, however, for us to consider here a
mechanical illustration of the rotation of the plane of polarized light by
magnetism which Thomson elaborated in 1874. This phenomenon may, I think,
be regarded as the most important of all Faraday's discoveries. It was of
enormous scientific importance, because it was made in a direction where a
new phenomenon was not even suspected. Of his discovery of induced currents
of electricity, to which all electric-lighting companies and transmission
of power companies of the present day owe their being, Faraday himself said
that it was a natural consequence of the discoveries of an earlier
experimenter, Oersted. But this magneto-optic discovery was quite
unexpected. I will now describe the phenomenon.

Some of you are aware that when a beam of light is sent through this
implement, called a Nichol's Prism, it becomes polarized, or
one-sided--that is, all the light that comes through is known to be
propagated by vibrations which occur all in one plane. This rope (Fig. 53)
hanging from the ceiling {119} illustrates the nature of plane polarized
light. All points in the rope are vibrating in the same plane. Well, this
prism A, Fig. 54, only lets through it light that is polarized in a
vertical plane. And here at B I have a similar implement, and I place it so
that it also will only allow light to pass through it which is polarized in
a vertical plane. Hence most of the light coming through the polarizer, as
the first prism is called, will pass readily through the analyzer, as the
second is called, and I am now letting this light enter my eye. But when I
turn the analyzer round through a right angle, I find that I see no light;
there was a gradual darkening as I rotated the analyzer. The analyzer will
now only allow light to pass through which is polarized in a horizontal
plane, and it receives no such light.

[Illustration: FIG. 53.]

[Illustration: FIG. 54.]

You will see in this model (Fig. 55) a good illustration of polarized
light. The white, brilliantly illuminated thread M N is {120} pulled by a
weight beyond the pulley M, and its end N is fastened to one limb of a
tuning-fork. Some ragged-looking pieces of thread round the portion N A
prevent its vibrating in any very determinate way, but from A to M the
thread is free from all encumbrance. A vertical slot at A, through which
the thread passes, determines the nature of the vibration of the part A B;
every part of the thread between A and B is vibrating in up and down
directions only. A vertical slot in B allows the vertical vibration to be
communicated through it, and so we see the part B M vibrating in the same
way as A B. I might point out quite a lot of ways in which this is not a
perfect illustration of what occurs with light in Fig. 54. But it is quite
good enough for my present purpose. A is a polarizer of vibration; it only
allows up and down motion to pass through it, and B also allows up and down
motion to pass through. But now, as B is turned round, it lets less and
less of the up and down motion pass through it, until when it is in the
second position shown in the lower part of the figure, it allows no up and
down motion to pass through, and there is no visible motion of the thread
between B and M. You will observe that if we did not know in what plane (in
the present case the plane is vertical) the vibrations of the thread
between A and B occurred, we should only have to turn B round until we
found no vibration {122} passing through, to obtain the information. Hence,
as in the light case, we may call A a polarizer of vibrations, and B an
analyzer.

[Illustration: FIG. 55.]

Now if polarized light is passing from A to B (Fig. 54) through the air,
say, and we have the analyzer placed so that there is darkness, we find
that if we place in the path of the ray some solution of sugar we shall no
longer have darkness at B; we must turn B round to get things dark again;
this is evidence of the sugar solution having twisted round the plane of
polarization of the light. I will now assume that you know something about
what is meant by twisting the plane of polarization of light. You know that
sugar solution will do it, and the longer the path of the ray through the
sugar, the more twist it gets. This phenomenon is taken advantage of in the
sugar industries, to find the strengths of sugar solutions. For the thread
illustration I am indebted to Professor Silvanus Thomson, and the next
piece of apparatus which I shall show also belongs to him.

I have here (_see_ Frontispiece) a powerful armour-clad coil, or
electro-magnet. There is a central hole through it, through which a beam of
light may be passed from an electric lamp, and I have a piece of Faraday's
heavy glass nearly filling this hole. I have a polarizer at one end, and an
analyzer at the other. You see now that the {123} polarized light passes
through the heavy glass and the analyzer, and enters the eye of an
observer. I will now turn B until the light no longer passes. Until now
there has been no magnetism, but I have the means here of producing a most
intense magnetic field in the direction in which the ray passes, and if
your eye were here you would see that there is light passing through the
analyzer. The magnetism has done something to the light, it has made it
capable of passing where it could not pass before. When I turn the analyzer
a little I stop the light again, and now I know that what the magnetism did
was to convert the glass into a medium like the sugar, a medium which
rotates the plane of polarization of light.

In this experiment you have had to rely upon my personal measurement of the
actual rotation produced. But if I insert between the polarizer and
analyzer this disc of Professor Silvanus Thomson's, built up of twenty-four
radial pieces of mica, I shall have a means of showing to this audience the
actual rotation of the plane of polarization of light. You see now on the
screen the light which has passed through the analyzer in the form of a
cross, and if the cross rotates it is a sign of the rotation of the plane
of polarization of the light. By means of this electric key I can create,
destroy, and reverse the magnetic {124} field in the glass. As I create
magnetism you see the twisting of the cross; I destroy the magnetism, and
it returns to its old position; I create the opposite kind of magnetism,
and you see that the cross twists in the opposite way. I hope it is now
known to you that magnetism rotates the plane of polarization of light as
the solution of sugar did.

[Illustration: FIG. 56.]

[Illustration: FIG. 57.]

As an illustration of what occurs between polarizer and analyzer, look
again at this rope (Fig. 53) fastened to the ceiling. I move the bottom end
sharply from east to west, and you see that every part of the rope moves
from east to west. Can you imagine a rope such that when the bottom end was
moved from east to west, a point some yards up moved from east-north-east
to west-sou'-west, that a higher point moved from north-east to south-west,
and so on, the direction gradually changing for higher and higher points?
Some of you, knowing what I have done, may be able to imagine it. We should
have what we want if this rope were a chain of gyrostats such as you see
figured in the diagram; gyrostats all spinning in the same way looked at
from below, with frictionless hinges between them. Here is such a chain
(Fig. 56), one of many that I have tried to use in this way for several
years. But although I have often believed that I saw the phenomenon occur
in {126} such a chain, I must now confess to repeated failures. The
difficulties I have met with are almost altogether mechanical ones. You see
that by touching all the gyrostats in succession with this rapidly
revolving disc driven by the little electromotor, I can get them all to
spin at the same time; but you will notice that what with bad mechanism and
bad calculation on my part, and want of skill, the phenomenon is completely
masked by wild movements of the gyrostats, the causes of which are better
known than capable of rectification. The principle of the action is very
visible in this gyrostat suspended as the bob of a pendulum (Fig. 57). You
may imagine this to represent a particle of the {127} substance which
transmits light in the magnetic field, and you see by the trickling thin
stream of sand which falls from it on the paper that it is continually
changing the plane of polarization. But I am happy to say that I can show
you to-night a really successful illustration of Thomson's principle; it is
the very first time that this most suggestive experiment has been shown to
an audience. I have a number of double gyrostats (Fig. 58) placed on the
same line, joined end to end by short pieces of elastic. Each instrument is
supported at its centre of gravity, and it can rotate both in horizontal
and in vertical planes.

[Illustration: FIG. 58.]

The end of the vibrating lever A can only get a horizontal motion from my
hand, and the motion is transmitted from one gyrostat to the next, until it
has travelled to the very end one. Observe that when the gyrostats are not
spinning, the motion is {128} everywhere horizontal. Now it is very
important not to have any illustration here of a reflected ray of light,
and so I have introduced a good deal of friction at all the supports. I
will now spin all the gyrostats, and you will observe that when A moves
nearly straight horizontally, the next gyrostat moves straight but in a
slightly different plane, the second gyrostat moves in another plane, and
so on, each gyrostat slightly twisting the plane in which the motion
occurs; and you see that the end one does not by any means receive the
horizontal motion of A, but a motion nearly vertical. This is a mechanical
illustration, the first successful one I have made after many trials, of
the effect on light of magnetism. The reason for the action that occurs in
this model must be known to everybody who has tried to follow me from the
beginning of the lecture.

And you can all see that we have only to imagine that many particles of the
glass are rotating like gyrostats, and that magnetism has partially caused
an allineation of their axes, to have a dynamical theory of Faraday's
discovery. The magnet twists the plane of polarization, and so does the
solution of sugar; but it is found by experiment that the magnet does it
indifferently for coming and going, whereas the sugar does it in a way that
corresponds with a spiral structure of molecules. You see that in this
important {129} particular the gyrostat analogue must follow the magnetic
method, and not the sugar method. We must regard this model, then, the
analogue to Faraday's experiment, as giving great support to the idea that
magnetism consists of rotation.

I have already exceeded the limits of time usually allowed to a popular
lecturer, but you see that I am very far from having exhausted our subject.
I am not quite sure that I have accomplished the object with which I set
out. My object was, starting from the very different behaviour of a top
when spinning and when not spinning, to show you that the observation of
that very common phenomenon, and a determination to understand it, might
lead us to understand very much more complex-looking things. There is no
lesson which it is more important to learn than this--That it is in the
study of every-day facts that all the great discoveries of the future lie.
Three thousand years ago spinning tops were common, but people never
studied them. Three thousand years ago people boiled water and made steam,
but the steam-engine was unknown to them. They had charcoal and saltpetre
and sulphur, but they knew nothing of gunpowder. They saw fossils in rocks,
but the wonders of geology were unstudied by them. They had bits of iron
and copper, but not one of them thought of any one of the fifty simple
{130} ways that are now known to us of combining those known things into a
telephone. Why, even the simplest kind of signalling by flags or lanterns
was unknown to them, and yet a knowledge of this might have changed the
fate of the world on one of the great days of battle that we read about. We
look on Nature now in an utterly different way, with a great deal more
knowledge, with a great deal more reverence, and with much less unreasoning
superstitious fear. And what we are to the people of three thousand years
ago, so will be the people of one hundred years hence to us; for indeed the
acceleration of the rate of progress in science is itself accelerating. The
army of scientific workers gets larger and larger every day, and it is my
belief that every unit of the population will be a scientific worker before
long. And so we are gradually making time and space yield to us and obey
us. But just think of it! Of all the discoveries of the next hundred years;
the things that are unknown to us, but which will be so well known to our
descendants that they will sneer at us as utterly ignorant, because these
things will seem to them such self-evident facts; I say, of all these
things, if one of us to-morrow discovered one of them, he would be regarded
as a great discoverer. And yet the children of a hundred years hence will
know it: it will be brought home to {131} them perhaps at every footfall,
at the flapping of every coat-tail.

Imagine the following question set in a school examination paper of 2090
A.D.--"Can you account for the crass ignorance of our forefathers in not
being able to see from England what their friends were doing in
Australia?"[13] Or this--"Messages are being received every minute from our
friends on the planet Mars, and are now being answered: how do you account
for our ancestors being utterly ignorant that these messages were
occasionally sent to them?" Or this--"What metal is as strong compared with
steel as steel is compared with lead? and explain why the discovery of it
was not made in Sheffield."

But there is one question that our descendants will never ask in accents of
jocularity, for to their bitter sorrow every man, woman, and child of them
will know the answer, and that question is this--"If our ancestors in the
matter of coal economy were not quite as ignorant as a baby who takes a
penny {132} as equivalent for a half-crown, why did they waste our coal?
Why did they destroy what never can be replaced?"

My friends, let me conclude by impressing upon you the value of knowledge,
and the importance of using every opportunity within your reach to increase
your own store of it. Many are the glittering things that seem to compete
successfully with it, and to exercise a stronger fascination over human
hearts. Wealth and rank, fashion and luxury, power and fame--these fire the
ambitions of men, and attract myriads of eager worshippers; but, believe
it, they are but poor things in comparison with knowledge, and have no such
pure satisfactions to give as those which it is able to bestow. There is no
evil thing under the sun which knowledge, when wielded by an earnest and
rightly directed will, may not help to purge out and destroy; and there is
no man or woman born into this world who has not been given the capacity,
not merely to gather in knowledge for his own improvement and delight, but
even to add something, however little, to that general stock of knowledge
which is the world's best wealth.

       *       *       *       *       *


{133}

ARGUMENT.

    1. _Introduction_, pages 9-14, showing the importance of the study of
    spinning-top behaviour.

    2. _Quasi-rigidity induced even in flexible and fluid bodies by rapid
    motion_, 14-21.

    Illustrations: Top, 14; belt or rope, 14; disc of thin paper, 14; ring
    of chain, 15; soft hat, 16; drunken man, 16; rotating water, 16; smoke
    rings, 17; Thomson's Molecular Theory, 19; swimmer caught in an eddy,
    20; mining water jet, 20; cased gyrostat, 21.

    3. _The nature of this quasi-rigidity in spinning bodies is a
    resistance to change of direction of the axis of spinning_, 21-30.

    Illustrations: Cased gyrostat, 21-24; tops, biscuits, hats, thrown into
    the air, 24-26; quoits, hoops, projectiles from guns, 27; jugglers at
    the Victoria Music Hall, 26-30; child trundling hoop, man on bicycle,
    ballet-dancer, the earth pointing to pole star, boy's top, 30.

    4. _Study of the crab-like behaviour of a spinning body_, 30-49.

    Illustrations: Spinning top, 31; cased gyrostat, 32; balanced gyrostat,
    33-36; windage of projectiles from {134} rifled guns, 36-38; tilting a
    hoop or bicycle, turning quickly on horseback, 38; bowls, 39; how to
    simplify one's observations, 39, 40; the illustration which gives us
    our simple universal rule, 40-42; testing the rule, 42-44; explanation
    of precession of gyrostat, 44, 45; precession of common top, 46;
    precession of overhung top, 46; list of our results given in a wall
    sheet, 48, 49.

    5. _Proof or explanation of our simple universal rule_, 50-54.

    Giving two independent rotations to a body, 50, 51; composition of
    rotations, 52, 53.

    6. _Warning that the rule is not, after all, so simple_, 54-66.

    Two independent spins given to the earth, 54; centrifugal force, 55;
    balancing of quick speed machinery, 56, 57; the possible wobbling of
    the earth, 58; the three principal axes of a body, 59; the free
    spinning of discs, cones, rods, rings of chain, 60; nodding motion of a
    gyrostat, 62; of a top, 63; parenthesis about inaccuracy of statement
    and Rankine's rhyme, 63, 64; further complications in gyrostatic
    behaviour, 64; strange elastic, jelly-like behaviour, 65; gyrostat on
    stilts, 66.

    7. _Why a gyrostat falls_, 66, 67.

    8. _Why a top rises_, 67-74.

    General ignorance, 67; Thomson preparing for the mathematical tripos,
    68; behaviour of a water-worn stone when spun on a table, 68, 69;
    parenthesis on technical education, 70; simple explanation of why a top
    rises, 70-73; behaviour of heterogeneous sphere when spun, 74.

    9. _Precessional motion of the earth_, 74-91.

    Its nature and effects on climate, 75-80; resemblance of the precessing
    earth to certain models, 80-82; tilting forces exerted by the sun and
    moon on the {135} earth, 82-84; how the earth's precessional motion is
    always altering, 85-88; the retrogression of the moon's nodes is itself
    another example, 88, 89; an exact statement made and a sort of apology
    for making it, 90, 91.

    10. _Influence of possible internal fluidity of the earth on its
    precessional motion_, 91-98.

    Effect of fluids and sand in tumblers, 91-93; three tests of the
    internal rigidity of an egg, that is, of its being a boiled egg, 93,
    94; quasi-rigidity of fluids due to rapid motion, forgotten in original
    argument, 95; beautiful behaviour of hollow top filled with water, 95;
    striking contrasts in the behaviour of two tops which are very much
    alike, 97, 98; fourth test of a boiled egg, 98.

    11. Apology for dwelling further upon astronomical matters, and
    impertinent remarks about astronomers, 99-101.

    12. How a gyrostat would enable a person living in subterranean regions
    to know, _1st, that the earth rotates_; _2nd, the amount of rotation_;
    _3rd, the direction of true north_; _4th, the latitude_, 101-111.

    Some men's want of faith, 101; disbelief in the earth's rotation, 102;
    how a free gyrostat behaves, 103, 104; Foucault's laboratory
    measurement of the earth's rotation, 105-107; to find the true north,
    108; all rotating bodies vainly endeavouring to point to the pole star,
    108; to find the latitude, 110; analogies between the gyrostat and the
    mariner's compass and the dipping needle, 110, 111; dynamical
    connection between magnetism and gyrostatic phenomena, 111.

    13. How the lecturer spun his tops, using electro-motors, 112-114.

    14. _Light_, _magnetism_, _and molecular spinning tops_, 115-128.

    Light takes time to travel, 115; the electro-magnetic {136} theory of
    light, 116, 117; signalling through fogs and buildings by means of a
    new kind of radiation, 117; Faraday's rotation of the plane of
    polarization by magnetism, with illustrations and models, 118-124;
    chain of gyrostats, 124; gyrostat as a pendulum bob, 126; Thomson's
    mechanical illustration of Faraday's experiment, 127, 128.

    15. _Conclusion_, 129-132.

    The necessity for cultivating the observation, 129; future discovery,
    130; questions to be asked one hundred years hence, 131; knowledge the
    thing most to be wished for, 132.

       *       *       *       *       *


{137}

APPENDIX I.

THE USE OF GYROSTATS.

In 1874 two famous men made a great mistake in endeavouring to prevent or
diminish the rolling motion of the saloon of a vessel by using a rapidly
rotating wheel. Mr. Macfarlane Gray pointed out their mistake. It is only
when the wheel is allowed to _precess_ that it can exercise a steadying
effect; the moment which it then exerts is equal to the angular speed of
the precession multiplied by the moment of momentum of the spinning wheel.

It is astonishing how many engineers who know the laws of motion of mere
translation, are ignorant of angular motion, and yet the analogies between
the two sets of laws are perfectly simple. I have set out these analogies
in my book on _Applied Mechanics_. The last of them between centripetal
force on a body moving in a curved path, and torque or moment on a rotating
body is the simple key to all gyrostatic or top calculation. When the spin
of a top is greatly reduced it is necessary to remember that the total
moment of momentum is not about the spinning axis (see my _Applied
Mechanics_, page 594); correction for this is, I suppose, what introduces
the complexity which scares students from studying the vagaries of tops;
but in all cases that are likely to come before an engineer it would be
absurd to study {138} such a small correction, and consequently calculation
is exceedingly simple.

Inventors using gyrostats have succeeded in doing the following things--

(1) Keeping the platform of a gun level on board ship, however the ship may
roll or pitch. Keeping a submarine vessel or a flying machine with any
plane exactly horizontal or inclined in any specified way.[14] It is easy
to effect such objects without the use of a gyrostat, as by means of spirit
levels it is possible to command powerful electric or other motors to keep
anything always level. The actual methods employed by Mr. Beauchamp Tower
(an hydraulic method), and by myself (an electric method), depend upon the
use of a gyrostat, which is really a pendulum, the axis being vertical.

(2) Greatly reducing the rolling (or pitching) of a ship, or the saloon of
a ship. This is the problem which Mr. Schlick has solved with great
success, at any rate in the case of torpedo boats.

(3) In Mr. Brennan's Mono-rail railway, keeping the resultant force due to
weight, wind pressure, centrifugal force, etc., exactly in line with the
rail, so that, however the load on a wagon may alter in position, and
although the wagon may be going round a curve, it is quickly brought to a
position such that there are no forces tending to alter its angular
position. The wagon leans over towards the wind or towards the centre of
the curve of the rail so as to be in equilibrium.

(4) I need not refer to such matters as the use of gyrostats for the
correction of compasses on board ship, referred to in page 111.

{139}

[Illustration: FIG. 1.]

{140} Problems (2) and (3) are those to which I wish to refer. For a ship
of 6,000 tons Mr. Schlick would use a large wheel of 10 to 20 tons,
revolving about an axis E F (fig. 1) whose mean position is vertical. Its
bearings are in a frame E C F D which can move about a thwart-ship axis C D
with a precessional motion. Its centre of gravity is below this axis, so
that like the ship itself the frame is in stable equilibrium. Let the ship
have rolled through an angle R from its upright position, and suppose the
axis E F to have precessed through the angle P from a vertical position.
Let the angular velocity of rolling be called [.R], and the angular
velocity of precession [.P]; let the moment of momentum of the wheel be m.
For any vibrating body like a ship it is easy to write out the equation of
motion; into this equation we have merely to introduce the moment m [.P]
diminishing R; into the equation for P we merely introduce the moment m
[.R] increasing P. As usual we introduce frictional terms; in the first
place F [.R] (F being a constant co-efficient) stilling the roll of the
ship; in the second case f [.P] a fluid friction introduced by a pair of
dash pots applied at the pins A and B to still the precessional vibrations
of the frame. It will be found that the angular motion P is very much
greater than the roll R. Indeed, so great is P that there are stops to
prevent its exceeding a certain amount. Of course so long as a stop acts,
preventing precession, the roll of the ship proceeds as if the gyrostat
wheel were not rotating. Mr. Schlick drives his wheels by steam; he will
probably in future do as Mr. Brennan does, drive them by electromotors, and
keep them in air-tight cases in good vacuums, because the loss of energy by
friction against an atmosphere is proportional to the density of the
atmosphere. The solution of the equations to find the nature of the R and P
motions is sometimes tedious, but requires no great amount of mathematical
knowledge. In a case considered by me of {141} a 6,000 ton ship, the period
of a roll was increased from 14 to 20 seconds by the use of the gyrostat,
and the roll rapidly diminished in amount. There was accompanying this slow
periodic motion, one of a two seconds' period, but if it did appear it was
damped out with great rapidity. Of course it is assumed that, by the use of
bilge keels and rolling chambers, and as low a metacentre as is allowable,
we have already lengthened the time of vibration, and damped the roll R as
much as possible, before applying the gyrostat. I take it that everybody
knows the importance of lengthening the period of the natural roll of a
ship, although he may not know the reason. The reason why modern ships of
great tonnage are so steady is because their natural periodic times of
rolling vibration are so much greater than the probable periods of any
waves of the sea, for if a series of waves acts upon a ship tending to make
it roll, if the periodic time of each wave is not very different from the
natural periodic time of vibration of the ship, the rolling motion may
become dangerously great.

If we try to apply Mr. Schlick's method to Mr. Brennan's car it is easy to
show that there is instability of motion, whether there is or is not
friction. If there is no friction, and we make the gyrostat frame unstable
by keeping its centre of gravity above the axis C D, there will be
vibrations, but the smallest amount of friction will cause these vibrations
to get greater and greater. Even without friction there will be instability
if m, the moment of momentum of the wheel, is less than a certain amount.
We see, then, that no form of the Schlick method, or modification of it,
can be applied to solve the Brennan problem.

{142}

[Illustration: FIG. 2.]

{143} Mr. Brennan's method of working is quite different from that of Mr.
Schlick. Fig. 2 shows his model car (about six feet long); it is driven by
electric accumulators carried by the car. His gyrostat wheels are driven by
electromotors (not shown in fig. 3); as they are revolving in nearly
vacuous spaces they consume but little power, and even if the current were
stopped they would continue running at sufficiently high speeds to be
effective for a length of time. Still it must not be forgotten that energy
is wasted in friction, and work has to be done in bringing the car to a new
position of equilibrium, and this energy is supplied by the electromotors.
Should the gyrostats really stop, or fall to a certain low speed, two
supports are automatically dropped, one on either side of the car; each of
them drops till it reaches the ground; one of them dropping, perhaps, much
farther than the other.

The real full-size car, which he is now constructing, may be pulled with
other cars by any kind of locomotive using electricity or petrol or steam,
or each of the wheels may be a driving wheel. He would prefer to generate
electropower on his train, and to drive every wheel with an electric motor.
His wheels are so independent of one another that they can take very quick
curves and vertical inequalities of the rail. The rail is fastened to
sleepers lying on ground that may have sidelong slope. The model car is
supported by a mono-rail bogie at each end; each bogie has two wheels
pivoted both vertically and horizontally; it runs on a round iron gas pipe,
and sometimes on steel wire rope; the ground is nowhere levelled or cut,
and at one place the rail is a steel wire rope spanning a gorge, as shown
in fig. 2. It is interesting to stop the car in the middle of this rope and
to swing the rope sideways to see the automatic balancing of the car. The
car may be left here or elsewhere balancing itself with nobody in charge of
it. If the load on the car--great lead weights--be dumped about into new
positions, the car adjusts itself to the new conditions with great {144}
quickness. When the car is stopped, if a person standing on the ground
pushes the car sidewise, the car of course pushes in opposition, like an
indignant animal, and by judicious pushing and yielding it is possible to
cause a considerable tilt. Left now to itself the car rights itself very
quickly.

[Illustration: FIG. 3.]

{145}

[Illustration: FIG. _3^b_ (showing the ground-plan of Fig. 3).]

{146} Fig. 3 is a diagrammatic representation of Mr. Brennan's pair of
gyrostats in sectional elevation and plan. The cases G and G', inside which
the wheels F and F' are rotating _in vacuo_ at the same speed and in
opposite directions (driven by electromotors not shown in the figure), are
pivoted about vertical axes E J and E' J'. They are connected by
spur-toothed segments J J and J' J', so that their precessional motions are
equal and opposite. The whole system is pivoted about C, a longitudinal
axis. Thus when precessing so that H comes out of the paper, so will H',
and when H goes into the paper, so does H'. When the car is in equilibrium
the axes K H and K' H' are in line N O O' N' across the car in the plane of
the paper. They are also in a line which is at right angles to the total
resultant (vertical or nearly vertical) force on the car. I will call
N O O' N' the mid position. Let ½m be the moment of momentum of either
wheel. Let us suppose that suddenly the car finds that it is not in
equilibrium because of a gust of wind, or centrifugal force, or an
alteration of loading, so that the shelf D comes up against H, the spinning
axis (or a roller revolving with the spinning axis) of the gyrostat. H
begins to roll away from me, and if no slipping occurred (but there always
is slipping, and, indeed, slipping is a necessary condition) it would roll,
that is, the gyrostats would precess with a constant angular velocity
[alpha], and exert the moment m[alpha] upon the shelf D, and therefore on
the car. It is to be observed that this is greater as the diameter of the
rolling part is greater. This precession continues until the roller and the
shelf cease to touch. At first H lifts with the shelf, and afterwards the
shelf moving downwards is followed for some distance by the roller. If the
tilt had been in the opposite direction the shelf D' would have acted
upwards upon the roller H', and caused just the opposite kind of
precession, and a moment of the opposite kind.

We now have the spindles out of their mid position; how are they brought
back from O Q and O' Q' to O N and O' N', {147} but with H permanently
lowered just the right amount? It is the essence of Mr. Brennan's invention
that after a restoring moment has been applied to the car the spindles
shall go back to the position N O O' N' (with H permanently lowered), so as
to be ready to act again. He effects this object in various ways. Some ways
described in his patents are quite different from what is used on the
model, and the method to be used on the full-size wagon will again be quite
different. I will describe one of the methods. Mr. Brennan tells me that he
considers this old method to be crude, but he is naturally unwilling to
allow me to publish his latest method.

D' is a circular shelf extending from the mid position in my direction; D
is a similar shelf extending from the mid position into the paper, or away
from me. It is on these shelves that H' and H roll, causing precession away
from N O O' N', as I have just described. When H' is inside the paper, or
when H is outside the paper, they find no shelf to roll upon. There are,
however, two other shelves L and L', for two other rollers M and M', which
are attached to the frames concentric with the spindles; they are free to
rotate, but are not rotated by the spindles. When they are pressed by their
shelves L or L' this causes negative precession, and they roll towards the
N O O' N' position. There is, of course, friction at their supports,
retarding their rotation, and therefore the precession. The important thing
to remember is that H and H', when they touch their shelves (when one is
touching the other is not touching) cause a precession away from the mid
position N O O' N' at a rate [alpha], which produces a restoring moment
m[alpha] of nearly constant amount (except for slipping), whereas where M
or M' touches its shelf L or L' (when one is touching the other is not
touching) the pressure on the shelf and friction determine the rate of the
precession towards the mid position N O O' N', {148} as well as the small
vertical motion. The friction at the supports of M and M' is necessary.

Suppose that the tilt from the equilibrium position to be corrected is R,
when D presses H upward. The moment m[alpha], and its time of action (the
total momental impulse) are too great, and R is over-corrected; this causes
the roller M' to act on L', and the spindles return to the mid position;
they go beyond the mid position, and now the roller H' acts on D', and
there is a return to the mid position, and beyond it a little, and so it
goes on, the swings of the gyrostats out of and into the mid position, and
the vibrations of the car about its position of equilibrium getting rapidly
less and less until again neither H nor H', nor M nor M' is touching a
shelf. It is indeed marvellous to see how rapidly the swings decay.
Friction accelerates the precession away from N O O' N'. Friction retards
the precession towards the middle position.

It will be seen that by using the two gyrostats instead of one when there
is a curve on the line, although the plane N O O' N' rotates, and we may
say that the gyrostats precess, the tilting couples which they might
exercise are equal and opposite. I do not know if Mr. Brennan has tried a
single gyrostat, the mid position of the axis of the wheel being vertical,
but even in this case a change of slope, or inequalities in the line, might
make it necessary to have a pair.

It is evident that this method of Mr. Brennan is altogether different in
character from that of Mr. Schlick. Work is here actually done which must
be supplied by the electromotors.

One of the most important things to know is this: the Brennan model is
wonderfully successful; the weight of the apparatus is not a large fraction
of the weight of the wagon; will this also be the case with a car weighing
1,000 times as {149} much? The calculation is not difficult, but I may not
give it here. If we assume that suddenly the wagon finds itself at the
angle R from its position of equilibrium, it may be taken that if the size
of each dimension of the wagon be multiplied by n, and the size of each
dimension of the apparatus be multiplied by p, then for a sudden gust of
wind, or suddenly coming on a curve, or a sudden shift of position of part
of the cargo, R may be taken as inversely proportional to n. I need not
state the reasonable assumption which underlies this calculation, but the
result is that if n is 10, p is 7.5. That is, if the weight of the wagon is
multiplied by 1,000, the weight of the apparatus is only multiplied by 420.
In fact, if, in the model, the weight of the apparatus is 10 per cent. of
that of the wagon, in the large wagon the weight of the apparatus is only
about 4 per cent. of that of the wagon. This is a very satisfactory
result.[15]

My calculations seem to show that Mr. Schlick's apparatus will form a
larger fraction of the whole weight of a ship, as the ship is larger, but
in the present experimental stage of the subject it is unfair to say more
than that this seems probable. My own opinion is that large ships are
sufficiently steady already.

In both cases it has to be remembered that if the _diameter_ of the wheel
can be increased in greater proportion than the dimensions of ship or
wagon, the proportional weight of the apparatus may be diminished. A wheel
of twice the diameter, but of the same weight, may have twice the moment of
momentum, and may therefore be twice as effective. I assume the stresses in
the material to be the same.

       *       *       *       *       *


{150}

APPENDIX II.

Page 23; note at line 3. Prof. Osborne Reynolds made the interesting remark
(_Collected Papers_, Vol. ii., p. 154), "That if solid matter had certain
kinds of internal motions, such as the box has, pears differing, say, from
apples, the laws of motion would not have been discovered; if discovered
for pears they would not have applied to apples."

Page 38; note at line 8. The motion of a rifle bullet is therefore one of
precession about the tangent to the path. The mathematical solution is
difficult, but Prof. Greenhill has satisfied himself mathematically that
air friction damps the precession, and causes the axis of the shot to get
nearer the tangential direction, so that fig. 10 illustrates what would
occur in a vacuum, but not in air. It is probable that this statement
applies only to certain proportions of length to diameter.

Page 129; note at line 5. Many men wonder how the ether can have the
enormous rigidity necessary for light transmission, and yet behave like a
frictionless fluid. One way of seeing how this may occur is to imagine that
when ordinary matter moves in the ether it only tends to produce motion of
translation of the ether particles, and therefore no resistance. But
anything such as light, which must operate in turning axes of rotating
parts, may encounter enormous elastic resistance.

_Richard Clay & Sons, Limited, London and Bungay._

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Notes

[1] The _Operatives' Lecture_ is always well advertised in the streets
beforehand by large posters.

[2] Bulwer Lytton's _Coming Race_.

[3] The glass vessel ought to be broader in comparison with its height.

[4] In 1746 Benjamin Robins taught the principles of rifling as we know
them now. He showed that the _spin_ of the round bullet was the most
important thing to consider. He showed that even the bent barrel of a gun
did not deflect the bullet to anything like the extent that the spin of the
bullet made it deflect in the opposite direction.

[5] NOTE.--In Fig. 16 the axis is shown inclined, but, only that it would
have been more troublesome to illustrate, I should have preferred to show
the precession occurring when the axis keeps horizontal.

[6] When this lecture containing the above statement was in the hands of
the printers, I was directed by Prof. Fitzgerald to the late Prof. Jellet's
_Treatise on the Theory of Friction_, published in 1872, and there at page
18 I found the mathematical explanation of the rising of a top.

[7] Roughly, the _Inertia_ or _Mass_ of a body expresses its resistance to
change of mere translational velocity, whereas, the _Moment of Inertia_ of
a body expresses its resistance to change of rotational velocity.

[8] It is a very unlikely, and certainly absurd-looking, hypothesis, but it
seems that it is not contradicted by any fact in spectrum analysis, or even
by any probable theory of the constitution of the interstellar ether, that
the stars are merely images of our own sun formed by reflection at the
boundaries of the ether.

[9] Sir William Thomson has performed this.

[10] It must be remembered that in one case I speak of the true north, and
in the other of the magnetic north.

[11] Rotating a large mass of iron rapidly in one direction and then in the
other in the neighbourhood of a delicately-suspended magnetic needle, well
protected from air currents, ought, I think, to give rise to magnetic
phenomena of very great interest in the theory of magnetism. I have
hitherto failed to obtain any trace of magnetic action, but I attribute my
failure to the comparatively slow speed of rotation which I have employed,
and to the want of delicacy of my magnetometer.

[12] I had applied for a patent for this system of signalling some time
before the above words were spoken, but although it was valid I allowed it
to lapse in pure shame that I should have so unblushingly patented the use
of the work of Fitzgerald, Hertz, and Lodge.

[13] How to see by electricity is perfectly well known, but no rich man
seems willing to sacrifice the few thousands of pounds which are necessary
for making the apparatus. If I could spare the money and time I would spend
them in doing this thing--that is, I think so--but it is just possible that
if I could afford to throw away three thousand pounds, I might feel greater
pleasure in the growth of a great fortune than in any other natural
process.

[14] Probably first described by Mr. Brennan.

[15] The weight of Mr. Brennan's loaded wagon is 313 lb., including
gyrostats and storage cells. His two wheels weigh 13 lb. If made of nickel
steel and run at their highest safe speed they would weigh much less.

       *       *       *       *       *


Changes made against printed original.

Page 91. "all that we should have to take into account": duplicated 'that'
in original.

Page 150. "applied to apples": 'applied to applies' in original.

Advertisements. "Persia ... by the Rev. Professor Sayce": 'Professsor' in
original.





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