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diff --git a/34268.txt b/34268.txt new file mode 100644 index 0000000..1058604 --- /dev/null +++ b/34268.txt @@ -0,0 +1,3447 @@ +The Project Gutenberg EBook of Spinning Tops, by John Perry + +This eBook is for the use of anyone anywhere at no cost and with +almost no restrictions whatsoever. You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Spinning Tops + +Author: John Perry + +Release Date: November 9, 2010 [EBook #34268] + +Language: English + +Character set encoding: ASCII + +*** START OF THIS PROJECT GUTENBERG EBOOK SPINNING TOPS *** + + + + +Produced by Chris Curnow, Keith Edkins and the Online +Distributed Proofreading Team at https://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + + + + +Transcriber's note: A few typographical errors have been corrected: they +are listed at the end of the text. + + * * * * * + + +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. + + * * * * * + + +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. + + * * * * * + + +{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 versa_. + +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 3-1/2 +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 23-1/2deg with the +ecliptic, which is the plane of the earth's orbit. Or the spinning axis of +the earth is always at angle of 23-1/2deg 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 23-1/2deg 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 +23-1/2deg. Under the influence of the attraction of this ring the earth +would heave like a great ship on a calm sea, rolling very slowly; in fact, +making one complete swing in about three years. But the earth is spinning, +and the tilting couple or torque acts upon it just like the forces which +are always tending to cause this ship-model to stand upright, and hence it +has a precessional motion whose complete period is 26,000 years. When there +is no spin in the ship, its complete oscillation takes place in three +seconds, and {85} when I spin the gyrostat on board the ship, the complete +period of its precession is two minutes. In both cases the effect of the +spin is to convert what would be an oscillation into a very much slower +precession. + +There is, however, a great difference between the earth and the gyrostat. +The forces acting on the top are always the same, but the forces acting on +the earth are continually altering. At midwinter and midsummer the tilting +forces are greatest, and at the equinoxes in spring and autumn there are no +such forces. So that the precessional motion changes its rate every quarter +year from a maximum to nothing, or from nothing to a maximum. It is, +however, always in the same direction--the direction opposed to the earth's +spin. When we speak then of the precessional motion of the earth, we +usually think of the mean or average motion, since the motion gets quicker +and slower every quarter year. + +Further, the moon is like the sun in its action. It tries to tilt the +equatorial part of the earth into the plane of the moon's orbit. The plane +of the moon's orbit is nearly the same as that of the ecliptic, and hence +the average precession of the earth is of much the same kind as if only one +of the two, the moon or the sun, alone acted. That is, the general +phenomenon of precession of the {86} earth's axis in a conical path in +26,000 years is the effect of the combined tilting actions of the sun and +moon. + +You will observe here an instance of the sort of untruth which it is almost +imperative to tell in explaining natural phenomena. Hitherto I had spoken +only of the sun as producing precession of the earth. This was convenient, +because the plane of the ecliptic makes always almost exactly 23-1/2deg +with the earth's equator, and although on the whole the moon's action is +nearly identical with that of the sun, and about twice as great, yet it +varies considerably. The superior tilting action of the moon, just like its +tide-producing action, is due to its being so much nearer us than the sun, +and exists in spite of the very small mass of the moon as compared with +that of the sun. + +As the ecliptic makes an angle of 23-1/2deg with the earth's equator, and +the moon's orbit makes an angle 5-1/2deg with the ecliptic, we see that the +moon's orbit sometimes makes an angle of 29deg with the earth's equator, +and sometimes only 18deg, changing from 29deg to 18deg, and back to 29deg +again in about nineteen years. This causes what is called "Nutation," or +the nodding of the earth, for the tilting action due to the sun is greatly +helped and greatly modified by it. The result of the variable nature of the +moon's action is then that the earth's axis {87} rotates in an elliptic +conical path round what might be called its mean position. We have also to +remember that twice in every lunar month the moon's tilting action on the +earth is greater, and twice it is zero, and that it continually varies in +value. + +On the whole, then, the moon and sun, and to a small extent the planets, +produce the general effect of a precession, which repeats itself in a +period of about 25,695 years. It is not perfectly uniform, being performed +at a speed which is a maximum in summer and winter; that is, there is a +change of speed whose period is half a year; and there is a change of speed +whose period is half a lunar month, the precession being quicker to-night +than it will be next Saturday, when it will increase for about another +week, and diminish the next. Besides this, because of 5-1/2deg of +angularity of the orbits, we have something like the nodding of our +precessing gyrostat, and the inclination of the earth's axis to the +ecliptic is not constant at 23-1/2deg, but is changing, its periodic time +being nineteen years. Regarding the earth's centre as fixed at O we see +then, as illustrated in this model and in Fig. 40, the axis of the earth +describes almost a perfect circle on the celestial sphere once in 25,866 +years, its speed fluctuating every half year and every half month. But it +is not a perfect circle, it is really a wavy {88} line, there being a +complete wave every nineteen years, and there are smaller ripples in it, +corresponding to the half-yearly and fortnightly periods. But the very +cause of the nutation, the nineteen-yearly period of retrogression of the +moon's nodes, as it is called, is itself really produced as the precession +of a gyrostat is produced, that is, by tilting forces acting on a spinning +body. + +[Illustration: FIG. 40.] + +Imagine the earth to be stationary, and the sun and moon revolving round +it. It was Gauss who found that the present action is the same as if the +masses of the moon and sun were distributed all {89} round their orbits. +For instance, imagine the moon's mass distributed over her orbit in the +form of a rigid ring of 480,000 miles diameter, and imagine less of it to +exist where the present speed is greater, so that the ring would be thicker +at the moon's apogee, and thinner at the perigee. Such a ring round the +earth would be similar to Saturn's rings, which have also a precession of +nodes, only Saturn's rings are not rigid, else there would be no +equilibrium. Now if we leave out of account the earth and imagine this ring +to exist by itself, and that its centre simply had a motion round the sun +in a year, since it makes an angle of 5-1/2deg with the ecliptic it would +vibrate into the ecliptic till it made the same angle on the other side and +back again. But it revolves once about its centre in twenty-seven solar +days, eight hours, and it will no longer swing like a ship in a +ground-swell, but will get a motion of precession opposed in direction to +its own revolution. As the ring's motion is against the hands of a watch, +looking from the north down on the ecliptic, this retrogression of the +moon's nodes is in the direction of the hands of a watch. It is exactly the +same sort of phenomenon as the precession of the equinoxes, only with a +much shorter period of 6798 days instead of 25,866 years. + +I told you how, if we knew the moon's mass or the sun's, we could tell the +amount of the forces, or {90} the torque as it is more properly called, +with which it tries to tilt the earth. We know the rate at which the earth +is spinning, and we have observed the precessional motion. Now when we +follow up the method which I have sketched already, we find that the +precessional velocity of a spinning body ought to be equal to the torque +divided by the spinning velocity and by the moment of inertia[7] of the +body about the polar axis. Hence the greater the tilting forces, and the +less the spin and the less the moment of inertia, the greater is the +precessional speed. Given all of these elements except one, it is easy to +calculate that unknown element. Usually what we aim at in such a +calculation is the determination of the moon's mass, as this phenomenon of +precession and the action of the tides are the only two natural phenomena +which have as yet enabled the moon's mass to be calculated. + +I do not mean to apologize to you for the introduction of such terms as +_Moment of Inertia_, nor do I mean to explain them. In this lecture I have +avoided, as much as I could, the introduction of mathematical expressions +and the use of technical terms. But I want you to {91} understand that I am +not afraid to introduce technical terms when giving a popular lecture. If +there is any offence in such a practice, it must, in my opinion, be greatly +aggravated by the addition of explanations of the precise meanings of such +terms. The use of a correct technical term serves several useful purposes. +First, it gives some satisfaction to the lecturer, as it enables him to +state, very concisely, something which satisfies his own weak inclination +to have his reasoning complete, but which he luckily has not time to +trouble his audience with. Second, it corrects the universal belief of all +popular audiences that they know everything now that can be said on the +subject. Third, it teaches everybody, including the lecturer, that there is +nothing lost and often a great deal gained by the adoption of a casual +method of skipping when one is working up a new subject. + +Some years ago it was argued that if the earth were a shell filled with +liquid, if this liquid were quite frictionless, then the moment of inertia +of the shell is all that we should have to take into account in considering +precession, and that if it were viscous the precession would very soon +disappear altogether. To illustrate the effect of the moment of inertia, I +have hung up here a number of glasses--one _a_ filled with sand, another +_b_ with treacle, a third _c_ with oil, the fourth _d_ with water, {92} + +[Illustration: FIG. 41.] + +{93} and the fifth _e_ is empty (Fig. 41). You see that if I twist these +suspending wires and release them, a vibratory motion is set up, just like +that of the balance of a watch. Observe that the glass with water vibrates +quickly, its effective moment of inertia being merely that of the glass +itself, and you see that the time of swing is pretty much the same as that +of the empty glass; that is, the water does not seem to move with the +glass. Observe that the vibration goes on for a fairly long time. + +The glass with sand vibrates slowly; here there is great moment of inertia, +as the sand and glass behave like one rigid body, and again the vibration +goes on for a long time. + +In the oil and treacle, however, there are longer periods of vibration than +in the case of the water or empty glass, and less than would be the case if +the vibrating bodies were all rigid, but the vibrations are stilled more +rapidly because of friction. + +Boiled (_f_) and unboiled (_g_) eggs suspended from wires in the same way +will exhibit the same differences in the behaviour of bodies, one of which +is rigid and the other liquid inside; you see how much slower an +oscillation the boiled has than the unboiled. + +Even on the table here it is easy to show the difference between boiled and +unboiled eggs. {94} Roll them both; you see that one of them stops much +sooner than the other; it is the unboiled one that stops sooner, because of +its internal friction. + +I must ask you to observe carefully the following very distinctive test of +whether an egg is boiled or not. I roll the egg or spin it, and then place +my finger on it just for an instant; long enough to stop the motion of the +shell. You see that the boiled egg had quite finished its motion, but the +unboiled egg's shell alone was stopped; the liquid inside goes on moving, +and now renews the motion of the shell when I take my finger away. + +It was argued that if the earth were fluid inside, the effective moment of +inertia of the shell being comparatively small, and having, as we see in +these examples, nothing whatever to do with the moment of inertia of the +liquid, the precessional motion of the earth ought to be enormously quicker +than it is. This was used as an argument against the idea of the earth's +being fluid inside. + +We know that the observed half-yearly and half-monthly changes of the +precession of the earth would be much greater than they are if the earth +were a rigid shell containing much liquid, and if the shell were not nearly +infinitely rigid the phenomena of the tides would not occur, but in regard +to the general precession of the earth there is now {95} no doubt that the +old line of argument was wrong. Even if the earth were liquid inside, it +spins so rapidly that it would behave like a rigid body in regard to such a +slow phenomenon as precession of the equinoxes. In fact, in the older line +of argument the important fact was lost sight of, that rapid rotation can +give to even liquids a quasi-rigidity. Now here (Fig. 42 _a_) is a hollow +brass top filled with water. The frame is light, and the water inside has +much more mass than the outside frame, and if you test this carefully you +will find that the top spins in almost exactly the same way as if the water +were quite rigid; in fact, as if the whole top were rigid. Here you see it +spinning and precessing just like any rigid top. This top, I know, is not +filled with water, it is only partially filled; but whether partially or +wholly filled it spins very much like a rigid top. + +[Illustration: FIG. 42.] + +{96} + +This is not the case with a long hollow brass top with water inside. I told +you that all bodies have one axis about which they prefer to rotate. The +outside metal part of a top behaves in a way that is now well known to you; +the friction of its peg on the table compels it to get up on its longer +axis. But the fluid inside a top is not constrained to spin on its longer +axis of figure, and as it prefers its shorter axis like all these bodies I +showed you, it spins in its own way, and by friction and pressure against +the case constrains the case to spin about the shorter axis, annulling +completely the tendency of the outside part to rise or keep up on its long +axis. Hence it is found to be simply impossible to spin a long hollow top +when filled with water. + +[Illustration: FIG. 43.] + +[Illustration: FIG. 44.] + +Here, for example, is one (Fig. 42 _b_) that only differs from the last in +being longer. It is filled, or partially filled, with water, and you +observe that if {97} I slowly get up a great spin when it is mounted in +this frame, and I let it out on the table as I did the other one, this one +lies down at once and refuses to spin on its peg. This difference of +behaviour is most remarkable in the two hollow tops you see before you +(Fig. 43). They are both nearly spherical, both filled with water. They +look so nearly alike that few persons among the audience are able to detect +any difference in their shape. But one of them (_a_) is really slightly +oblate like an orange, and the other (_b_) is slightly prolate like a +lemon. I will give them both a gradually increasing rotation in this frame +{98} (Fig. 44) for a time sufficient to insure the rotation of the water +inside. When just about to be set free to move like ordinary tops on the +table, water and brass are moving like the parts of a rigid top. You see +that the orange-shaped one continues to spin and precess, and gets itself +upright when disturbed, like an ordinary rigid top; indeed I have seldom +seen a better behaved top; whereas the lemon-shaped one lies down on its +side at once, and quickly ceases to move in any way. + +[Illustration: FIG. 45.] + +And now you will be able to appreciate a fourth test of a boiled egg, which +is much more easily seen by a large audience than the last. Here is the +unboiled one (Fig. 45 _b_). I try my best to spin it as it lies on the +table, but you see that I cannot give it much spin, and so there is nothing +of any importance to look at. But you observe that it is quite easy to spin +the boiled {99} egg, and that for reasons now well known to you it behaves +like the stones that Thomson spun on the sea-beach; it gets up on its +longer axis, a very pretty object for our educated eyes to look at (Fig. 45 +_a_). You are all aware, from the behaviour of the lemon-shaped top, that +even if, by the use of a whirling table suddenly stopped, or by any other +contrivance, I could get up a spin in this unboiled egg, it would never +make the slightest effort to rise on its end and spin about its longer +axis. + +I hope you don't think that I have been speaking too long about +astronomical matters, for there is one other important thing connected with +astronomy that I must speak of. You see, I have had almost nothing +practically to do with astronomy, and hence I have a strong interest in the +subject. It is very curious, but quite true, that men practically engaged +in any pursuit are almost unable to see the romance of it. This is what the +imaginative outsider sees. But the overworked astronomer has a different +point of view. As soon as it becomes one's duty to do a thing, and it is +part of one's every-day work, the thing loses a great deal of its interest. +We have been told by a great American philosopher that the only coachmen +who ever saw the romance of coach-driving are those titled individuals who +pay nowadays so largely for the {100} privilege. In almost any branch of +engineering you will find that if any invention is made it is made by an +outsider; by some one who comes to the study of the subject with a fresh +mind. Who ever heard of an old inhabitant of Japan or Peru writing an +interesting book about those countries? At the end of two years' residence +he sees only the most familiar things when he takes his walks abroad, and +he feels unmitigated contempt for the ingenuous globe-trotter who writes a +book about the country after a month's travel over the most beaten tracks +in it. Now the experienced astronomer has forgotten the difficulties of his +predecessors and the doubts of outsiders. It is a long time since he felt +that awe in gazing at a starry sky that we outsiders feel when we learn of +the sizes and distances apart of the hosts of heaven. He speaks quite +coolly of millions of years, and is nearly as callous when he refers to the +ancient history of humanity on our planet as a weather-beaten geologist. +The reason is obvious. Most of you know that the _Nautical Almanac_ is as a +literary production one of the most uninteresting works of reference in +existence. It is even more disconnected than a dictionary, and I should +think that preparing census-tables must be ever so much more romantic as an +occupation than preparing the tables of the _Nautical Almanac_. And yet +{101} a particular figure, one of millions set down by an overworked +calculator, may have all the tragic importance of life or death to the crew +and passengers of a ship, when it is heading for safety or heading for the +rocks under the mandate of that single printed character. + +But this may not be a fair sort of criticism. I so seldom deal with +astronomical matters, I know so little of the wear and tear and monotony of +the every-day life of the astronomer, that I do not even know that the +above facts are specially true about astronomers. I only know that they are +very likely to be true because they are true of other professional men. + +I am happy to say that I come in contact with all sorts and conditions of +men, and among others, with some men who deny many of the things taught in +our earliest school-books. For example, that the earth is round, or that +the earth revolves, or that Frenchmen speak a language different from ours. +Now no man who has been to sea will deny the roundness of the earth, +however greatly he may wonder at it; and no man who has been to France will +deny that the French language is different from ours; but many men who +learnt about the rotation of the earth in their school-days, and have had a +plentiful opportunity of observing the heavenly bodies, deny the rotation +of the earth. {102} They tell you that the stars and moon are revolving +about the earth, for they see them revolving night after night, and the sun +revolves about the earth, for they see it do so every day. And really if +you think of it, it is not so easy to prove the revolution of the earth. By +the help of good telescopes and the electric telegraph or good +chronometers, it is easy to show from the want of parallax in stars that +they must be very far away; but after all, we only know that either the +earth revolves or else the sky revolves.[8] Of course, it seems infinitely +more likely that the small earth should revolve than that the whole +heavenly host should turn about the earth as a centre, and infinite +likelihood is really absolute proof. Yet there is nobody who does not +welcome an independent kind of proof. The phenomena of the tides, and +nearly every new astronomical fact, may be said to be an addition to the +proof. Still there is the absence of perfect certainty, and when we are +told that these spinning-top phenomena give us a real proof of the rotation +of the earth without our leaving the room, we welcome {103} it, even +although we may sneer at it as unnecessary after we have obtained it. + +[Illustration: FIG. 17.] + +You know that a gyrostat suspended with perfect freedom about axes, which +all pass through its centre of gravity, maintains a constant direction in +space however its support may be carried. Its axis is not forced to alter +its direction in any way. Now this gyrostat (Fig. 17) has not the perfect +absence of friction at its axes of which I speak, and even the slightest +friction will produce some constraint which is injurious to the experiment +I am about to describe. It must be remembered, that if there were +absolutely no constraint, then, even if the {104} gyrostat were _not_ +spinning, its axis would keep a constant direction in space. But the +spinning gyrostat shows its superiority in this, that any constraint due to +friction is less powerful in altering the axis. The greater the spin, then, +the better able are we to disregard effects due to friction. You have seen +for yourselves the effect of carrying this gyrostat about in all sorts of +ways--first, when it is not spinning and friction causes quite a large +departure from constancy of direction of the axis; second, when it is +spinning, and you see that although there is now the same friction as +before, and I try to disturb the instrument more than before, the axis +remains sensibly parallel to itself all the time. Now when this instrument +is supported by the table it is really being carried round by the earth in +its daily rotation. If the axis kept its direction perfectly, and it were +now pointing horizontally due east, six hours after this it will point +towards the north, but inclining downwards, six hours afterwards it will +point due west horizontally, and after one revolution of the earth it will +again point as it does now. Suppose I try the experiment, and I see that it +points due east now in this room, and after a time it points due west, and +yet I know that the gyrostat is constantly pointing in the same direction +in space all the time, surely it is obvious that the room must {105} be +turning round in space. Suppose it points to the pole star now, in six +hours, or twelve, or eighteen, or twenty-four, it will still point to the +pole star. + +Now it is not easy to obtain so frictionless a gyrostat that it will +maintain a good spin for such a length of time as will enable the rotation +of the room to be made visible to an audience. But I will describe to you +how forty years ago it was proved in a laboratory that the earth turns on +its axis. This experiment is usually connected with the name of Foucault, +the same philosopher who with Fizeau showed how in a laboratory we can +measure the velocity of light, and therefore measure the distance of the +sun. It was suggested by Mr. Lang of Edinburgh in 1836, although only +carried out in 1852 by Foucault. By these experiments, if you were placed +on a body from which you could see no stars or other outside objects, say +that you were living in underground regions, you could discover--first, +whether there is a motion of rotation, and the amount of it; second, the +meridian line or the direction of the true north; third, your latitude. +Obtain a gyrostat like this (Fig. 46) but much larger, and far more +frictionlessly suspended, so that it is free to move vertically or +horizontally. For the vertical motion your gymbal pivots ought to be hard +steel knife-edges. {106} + +[Illustration: FIG. 46.] + +As for the horizontal freedom, Foucault used a fine steel wire. Let there +be a fine scale engraved crosswise on the outer gymbal ring, and try to +discover if it moves horizontally by means of a microscope with cross +wires. When this is carefully done we find that there is a motion, {107} +but this is not the motion of the gyrostat, it is the motion of the +microscope. In fact, the microscope and all other objects in the room are +going round the gyrostat frame. + +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 1/2m 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._ + + * * * * * + + +PUBLICATIONS + +OF THE + +SOCIETY FOR PROMOTING CHRISTIAN KNOWLEDGE. + +THE ROMANCE OF SCIENCE. + +Small post 8vo, cloth boards. + + COAL, AND WHAT WE GET FROM IT. By Professor R. MELDOLA, F.R.S., F.I.C. + With several Illustrations. 2s. 6d. + + COLOUR MEASUREMENT AND MIXTURE. By Sir W. DE W. ABNEY, K.C.B., R.E., + F.R.S. Numerous Illustrations. 2s. 6d. + + DISEASES OF PLANTS. By Professor MARSHALL WARD. Numerous Illustrations. + 2s. 6d. + + OUR SECRET FRIENDS AND FOES. By PERCY FARADAY FRANKLAND, Ph.D., B.Sc. + (Lond.), F.R.S. Illustrated. 3s. + + SOAP-BUBBLES, AND THE FORCES WHICH MOULD THEM. 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SYMONDS. _2s. 6d._ + + * * * * * + +LONDON: NORTHUMBERLAND AVENUE, W.C. +43 QUEEN VICTORIA STREET, E.C. + + * * * * * + + +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. + + + + + + +End of the Project Gutenberg EBook of Spinning Tops, by John Perry + +*** END OF THIS PROJECT GUTENBERG EBOOK SPINNING TOPS *** + +***** This file should be named 34268.txt or 34268.zip ***** +This and all associated files of various formats will be found in: + https://www.gutenberg.org/3/4/2/6/34268/ + +Produced by Chris Curnow, Keith Edkins and the Online +Distributed Proofreading Team at https://www.pgdp.net (This +file was produced from images generously made available +by The Internet Archive) + + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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