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[PDF page 77] CH. II] PROBLEMS WITH COUNTERS OR PA WNS. 61 direction; the “a” pieces can move from left to right, and the “b” pieces from right to left. If the square next to a piece is unoccupied, it can a a a b b b move on to that; or if the square next to it is occupied by a piece of the opposite colour and the square beyond that is unoccupied, then it can, like a queen in draughts, leap over that piece on to the unoccupied square beyond it. The object is to get all the white pawns in the places occupied initially by the black pawns and vice versa. The solution requires15moves. It may be effected by moving first a white pawn, then successively two black pawns then three white pawns, then three black pawns, then three white pawns, then two black pawns, and then one white pawn. We can express this solution by saying that if we number the cells (a term used to describe each of the small squares on a chess-board) consecutively, then initially the vacant space occupies the cell 4 and in the successive moves it will occupy the cells3, 5, 6, 4, 2, 1, 3, 5, 7, 6, 4, 2, 3, 5, 4. Of these moves, six are simple and nine are leaps. Similarly if we haven white pawns at one end of a row of2n + 1 cells, and n black pawns at the other end, they can be interchanged in n(n+2) moves, by moving in succession1 pawn, 2 pawns, 3 pawns, ..., n − 1 pawns, n pawns, n pawns, n pawns, n − 1 pawns, ..., 2 pawns, and 1 pawn—all the pawns in each group being of the same colour and different from that of the pawns in the group preceding it. Of these moves 2n are simple and n2 are leaps. Second Problem with Pawns*. A similar game may be played on a rectangular or square board. The case of a square board containing 49 cells, or small squares, will illustrate this sufficiently: in this case the initial position is shown in the annexed diagram where the “a”s denote the pawns or pieces of one colour, and the “b”s those of the other colour. The “a” pieces can move horizontally from left to right or vertically down, and the “b” pieces can move horizontally from right to left or vertically up, according to the same rules as before. The solution reduces to the preceding case. The pieces in the mid- dle column can be interchanged in15 moves. In the course of these moves every one of the seven cells in that column is at some time or * Lucas, vol. ii, part 5, p. 144.

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[PDF page 78] 62 SOME GEOMETRICAL QUESTIONS. [CH. II a a a a b b b a a a a b b b a a a a b b b a a a b b b a a a b b b b a a a b b b b a a a b b b b other vacant, and whenever that is the case the pieces in the row con- taining the vacant cell can be interchanged. To interchange the pieces in each of the seven rows will require 15 moves. Hence to interchange all the pieces will require15 + (7 × 15) moves, that is,120 moves. If we place2n(n + 1) white pawns and2n(n + 1) black pawns in a similar way on a square board of(2n + 1)2 cells, we can transpose them in 2n(n + 1)(n + 2) moves: of these 4n(n + 1) are simple and 2n2(n + 1) are leaps. Third Problem with Pawns. The following analogous, though somewhat more complicated, game was I believe originally published in the first edition of this work: but I find that it has been since widely a b c d e f g h ∗ H G F E D C B A distributed in connexion with an advertisement and probably now is well-known. On a square board of 25 cells, place eight white pawns or counters on the cells denoted by small letters in the annexed diagram, and eight black pawns or counters on the cells denoted by capital let- ters: the cell marked with an asterisk (∗) being left blank. Each pawn can move according to the laws already explained—the white pawns being able to move only horizontally from left to right or vertically downwards, and the black pawns being able to move only horizontally from right to left or vertically upwards. The object is to get all the

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[PDF page 79] CH. II] PROBLEMS WITH CHESS-PIECES 63 white pawns in the places initially occupied by the black pawns and vice versa. No moves outside the dark line are permitted. Since there is only one cell on the board which is unoccupied, and since no diagonal moves and no backward moves are permitted, it fol- lows that at each move not more than two pieces of either colour are capable of moving. There are however a very large number of solu- tions. The following empirical solution in forty-eight moves is one way of effecting the transfer—the letters indicating the cellsfrom which the pieces are successively moved: h H ∗ f F E H G ∗ c b h g d f F C ∗ h H B A C ∗ c a b h H ∗ c f F D G H B C ∗ g h e f F ∗ h H ∗ . It will be noticed that the first twenty-four moves lead to a symmet- rical position, and that the next twenty-three moves can be at once obtained by writing the first twenty-three moves in reverse order and interchanging small and capital letters. Probably, were it worth the trouble, the mathematical theory of games such as that just described might be worked out by the use of Vandermonde’s notation, described later in chaptervi, or by the analogous method employed in the theory of the game of solitaire*. I believe that this has not been done, and I do not think it would repay the labour involved. Problems on a Chess-board with Chess-pieces. There are several mathematical recreations with chess-pieces, other than pawns, some- what similar to those given above. One of these, on the determination of the ways in which eight queens can be placed on a board so that no queen can take any other, is given later in chapteriv. Another, on the path to be followed by a knight which is moved on a chess-board so that it shall occupy every cell once and only once, is given in chaptervi. Here I will mention one of the simplest of such problems, which is in- teresting from the fact that it is given in Guarini’s manuscript written in 1512; it was quoted by Lucas, but so far as I know has not been otherwise published. Guarini’s Problem.On a board of nine cells, such as that drawn below, the two white knights are placed on the two top corner cells * On the theory of the solitaire, see Reiss, ‘Beiträge zur Theorie des Solitär-Spiels,’ Crelle’s Journal, Berlin, 1858, vol.liv, pp. 344–379; and Lucas, vol.i, part v, pp. 89–141.

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[PDF page 80] 64 SOME GEOMETRICAL QUESTIONS. [CH. II a C d D B b A c (a, d), and the two black knights on the two bottom corner cells (b, c): the other cells are left vacant. It is required to move the knights so that the white knights shall occupy the cellsb and c, while the black shall occupy the cells a and d. The solution is tolerably obvious. First, move the pieces froma to A, from b to B, from c to C, and fromd to D. Next, move the pieces from A to d, from B to a, from C to b, and fromD to c. The effect of these eight moves is the same as if the original square had been rotated through one right angle. Repeat the above process, that is, move the pieces successively from a to A, from b to B, from c to C, from d to D; from A to d, from B to a, from C to b, and from D to c. The required result is then attained. Geometrical Puzzles with Rods, etc. Another species of geometrical puzzles, to which here I will do no more than allude, are made of steel rods, or of wire, or of wire and string. Numbers of these are often sold in the streets of London for a penny each, and some of them afford ingenious problems in the geometry of position. Most of them could hardly be discussed without the aid of diagrams, but they are inexpensive to construct, and in fact innumerable puzzles on geometry of position can be made with a couple of stout sticks and a ball of string, or even with only a box of matches: several examples are given in the appendix to the fourth volume of the 1723 edition of Ozanam’s work. I will mention, as an easy example, analogous to one group of the string puzzles, that any one can take off his waistcoat (which may be unbuttoned) without taking off his coat, and without pulling the waistcoat over the head like a jersey. This last feat may serve to show the difficulty of mentally realizing the effect of geometrical alterations in a figure unless they are of the simplest character. Paradromic Rings. The fact just stated is illustrated by the familiar experiment of making paradromic rings by cutting a paper

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[PDF page 81] CH. II] PARADROMIC RINGS. 65 ring prepared in the following manner. Take a strip of paper or piece of tape, say, for convenience, an inch or two wide and at least nine or ten inches long, rule a line in the middle down the lengthAB of the strip, gum one end over the other end B, and we get a ring like a section of a cylinder. If this ring is cut by a pair of scissors along the ruled line we obtain two rings exactly like the first, except that they are only half the width. Next suppose that the endA is twisted through two right angles before it is gummed to B (the result of which is that the back of the strip atA is gummed over the front of the strip atB), then a cut along the line will produce only one ring. Next suppose that the endA is twisted once completely round (i.e. through four right angles) before it is gummed toB, then a similar cut produces two interlaced rings. If any of my readers think that these results could be predicted off-hand, it may be interesting to them to see if they can predict correctly the effect of again cutting the rings formed in the second and third experiments down their middle lines in a manner similar to that above described. The theory is due to J.B. Listing* who discussed the case when the endA receives m half-twists, that is, is twisted throughmπ, before it is gummed to B. If m is even we obtain a surface which has two sides and two edges, which are termed paradromic. If the ring is cut along a line midway between the edges, we obtain two rings, each of which hasmhalf-twists, and which are linked together 1 2 m times. If m is odd we obtain a surface having only one side and one edge. If this ring is cut along its mid-line, we obtain only one ring, but it has 2m half-twists, and ifm is greater than unity it is knotted. * Vorstudien zur Topologie, Die Studien, Göttingen, 1847, partx.

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[PDF page 82] CHAPTER III. SOME MECHANICAL QUESTIONS. I proceednow to enumerate a few questions connected with me- chanics which lead to results that seem to me interesting from a his- torical point of view or paradoxical. Problems in mechanics generally involvemoredifficultiesthanproblemsinarithmetic, algebra, orgeome- try, and the explanations of some phenomena—such as those connected with the flight of birds—are still incomplete, while the explanations of many others of an interesting character are too difficult to find a place in a non-technical work. Here, however, I shall confine myself to ques- tions which, like those treated in the two preceding chapters, are of an elementary, not to say trivial, character; and the conclusions are well-known to mathematicians. I assume that the reader is acquainted with the fundamental ideas of kinematics and dynamics, and is familiar with the three Newtonian laws; namely, first that a body will continue in its state of rest or of uniform motion in a straight line unless compelled to change that state by some external force: second, that the change of momentum per unit of time is proportional to the external force and takes place in the direction of it: and third, that the action of one body on another is equal in magnitude but opposite in direction to the reaction of the second body on the first. The first and second laws state the principles required for solving any question on the motion of a particle under the action of given forces. The third law supplies the additional principle required for the solution of problems in which two or more particles influence one another. 66

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[PDF page 83] CH. III] ZENO’S PARADOXES.. 67 Motion. The difficulties connected with the idea of motion have been for a long time a favourite subject for paradoxes, some of which bring us into the realm of the philosophy of mathematics. Zeno’s Paradoxes on Motion.One of the earliest of these is the remark of Zeno to the effect that since an arrow cannot move where it is not, and since also it cannot move where it is (i.e. in the space it exactly fills), it follows that it cannot move at all. The answer that the very idea of the motion of the arrow implies the passage from where it is to where it is not was rejected by Zeno, who seems to have thought that the appearance of motion of a body was a phenomenon caused by the successive appearances of the body at rest but in different positions. Zeno also asserted that the idea of motion was itself inconceivable, for what moves must reach the middle of its course before it reaches the end. Hence the assumption of motion presupposes another motion, and that in turn another, and so ad infinitum. His objection was in fact analogous to the biological difficulty expressed by Swift:— “So naturalists observe, a flea hath smaller fleas that on him prey. And these have smaller fleas to bite ’em. And so proceed ad infinitum.” Or as De Morgan preferred to put it “Great fleas have little fleas upon their backs to bite ’em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these have greater still, and greater still, and so on.” Achilles and the Tortoise. Zeno’s paradox about Achilles and the tortoise is known even more widely. The assertion was that if Achilles ran ten times as fast as a tortoise, yet if the tortoise had (say)1000 yards start it could never be overtaken. To establish this, Zeno argued that when Achilles had gone the1000 yards, the tortoise would still be 100 yards in front of him; by the time he had covered these100 yards, it would still be 10 yards in front of him; and so on for ever. Thus Achilles would get nearer and nearer to the tortoise but would never overtake it. Zeno regarded this as confirming his view that the popular idea of motion is self-contradictory.

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[PDF page 84] 68 MECHANICAL RECREATIONS. [CH. III Zeno’s Paradox on Time.The fallacy of Achilles and the Tortoise is usually explained by saying that though the time required to over- take the tortoise can be divided into an infinite number of intervals, as stated in the argument, yet these intervals get smaller and smaller in geometrical progression, and the sum of them all is a finite time: after the lapse of that time Achilles would be in front of the tortoise. Probably Zeno would have replied that this explanation rests on the assumption that space and time are infinitely divisible, propositions which he would not admit. He seems further to have contended that while, to an accurate thinker, the notion of the infinite divisibility of time was impossible, it was equally impossible to think of a minimum measure of time. For suppose, he argued, thatτ is the smallest con- ceivable interval, and suppose that three horizontal lines composed of three consecutive spansabc, a′b′c′, a′′b′′c′′ are placed so thataa′a′′, bb′b′′, cc′c′′ are vertically over one another. Imagine the second line moved as a whole one span to the right in the timeτ, and simultaneously the third line moved as a whole one span to the left. Thenb, a′, c′′ will be vertically over one another. And in this durationτ (which by hypothesis is indivisible)c′ must have passed vertically overa′′. Hence the duration is divisible, contrary to the hypothesis. The Paradox of Tristram Shandy. Mr Russell has enunciated* a paradox somewhat similar to that of Achilles and the Tortoise, save that the intervals of time considered get longer and longer during the course of events. Tristram Shandy, as we know, took two years writing the history of the first two days of his life, and lamented that, at this rate, material would accumulate faster than he could deal with it, so that he could never come to an end, however long he lived. But had he lived long enough, and not wearied of his task, then, even if his life had continued as eventfully as it began, no part of his biography would have remained unwritten. For if he wrote the events of the first day in the first year, he would write the events of thenth day in thenth year, hence in time the events of any assigned day would be written, and therefore no part of his biography would remain unwritten. This argument might be put in the form of a demonstration that the part of a magnitude may be equal to the whole of it. Questions, such as those given above, which are concerned with the continuity and extent of space and time involve difficulties of a * B.A.W. Russell, Principles of Mathematics, Cambridge, 1903, vol.i, p. 358.

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[PDF page 85] CH. III] ANGULAR MOTION. 69 high order. Angular Motion. A non-mathematician finds additional difficul- ties in the idea of angular motion. For instance, here is a well-known proposition on motion in an equiangular spiral (of which the result is true on the ordinary conventions of mathematics) which shows that a body, moving with uniform velocity and as slowly as we please, may in a finite time whirl round a fixed point an infinite number of times. The equiangular spiral is the trace of a pointP, which moves along a line OP, the line OP turning round a fixed pointO with uniform angular velocity while the distance of P from O decreases with the time in geometrical progression. If the radius vector rotates through four right angles we have one convolution of the curve. All convolutions are similar, and the length of each convolution is a constant fraction, say 1/nth, that of the convolution immediately outside it. Inside any given convolution, there are an infinite number of convolutions which get smaller and smaller as we get nearer the pole. Now suppose a point Q to move uniformly along the spiral from any point towards the pole. If it covers the first convolution ina seconds, it will cover the next in a/n seconds, the next ina/n2 seconds, and so on, and will finally reach the pole in(a + a/n + a/n2 + a/n3 + · · ·) seconds, that is, inan/(n − 1) seconds. The velocity is uniform, and yet in a finite time,Q will have traversed an infinite number of convolutions and therefore have circled round the pole an infinite number of times*. Simple Relative Motion. Even if the philosophical difficulties sug- gested by Zeno are settled or evaded, the mere idea of relative motion has been often found to present difficulties, and Zeno himself failed to explain a simple phenomenon involving the principle. As one of the easiest examples of this kind, I may quote the common question of how many trains going fromB to A a passenger from A to B would meet and pass on his way, assuming that the journey either way takes4 1 2 hours and that the trains start from each end every hour. The answer is 9. Or again this: Take two pennies, face upwards on a table and edges in contact. Suppose that one is fixed and that the other rolls on it without slipping, making one complete revolution round it and returning to its initial position. How many revolutions round its own centre has the rolling coin made? The answer is2. * The proposition is put in this form in J. Richard’sPhilosophie des mathémat- iques, Paris, 1903, pp. 119–120.

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[PDF page 86] 70 MECHANICAL RECREATIONS. [CH. III Laws of Motion. I proceed next to make a few remarks on points connected with the laws of motion. The first law of motion is often said to defineforce, but it is in only a qualified sense that this is true. Probably the meaning of the law is best expressed in Clifford’s phrase, that force is “the description of a certain kind of motion”—in other words it is not an entity but merely a convenient way of stating, without circumlocution, that a certain kind of motion is observed. It is not difficult to show that any other interpretation lands us in difficulties. Thus some authors use the law to justify a definition that force is that which moves a body or changes its motion; yet the same writers speak of a steam-engine moving a train. It would seem then that, according to them, a steam-engine is a force. That such state- ments are current may be fairly reckoned among mechanical paradoxes. The idea of force is difficult to grasp. How many people, for in- stance, could predict correctly what would happen in a question as sim- ple as the following? A rope (whose weight may be neglected) hangs over a smooth pulley; it has one end fastened to a weight of10 stone, and the other end to a sailor of weight10 stone, the sailor and the weight hanging in the air. The sailor begins to climb up the rope; will the weight move at all; and, if so, will it rise or fall? It will be noted that in the first law of motion it is asserted that, unless acted on by an external force, a body in motion continues to move (i) with uniform velocity, and (ii) in a straight line. The tendency of a body to continue in its state of rest or of uniform motion is called its inertia. This tendency may be used to explain various common phenomena and experiments. Thus, if a number of dominoes or draughts are arranged in a vertical pile, a sharp horizontal blow on one of those near the bottom will send it out of the pile, and those above will merely drop down to take its place—in fact they have nottimetochangetheirrelativepositionsbeforethereissufficientspace for them to drop vertically as if they were a solid body. This also is the principle on which depends the successful playing of “Aunt Sally,” and the performance of numerous tricks, described in collections of mathematical puzzles*. * See Les récréations scientifiquesby G. Tissandier, where several ingenious illus- trations of inertia are given.

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[PDF page 87] CH. III] LA WS OF MOTION. 71 The statement about inertia in the first law may be taken to imply that a body set in rotation about a principal axis passing through its centre of mass will continue to move with a uniform angular velocity and to keep its axis of rotation fixed in direction. The former of these statements is the assumption on which our measurement of time is based as mentioned below in chapter xiii. The latter assists us to explain the motion of a projectile in a resisting fluid. It affords the explanation of why the barrel of a rifle is grooved; and why, similarly, anyone who has to throw a flat body of irregular shape (such as a card) in a given direction usually gives it a rapid rotatory motion about a principal axis. Elegant illustrations of the fact just mentioned are afforded by a good many of the tricks of acrobats, though the full explanation of most of them also introduces other considerations. Thus when some few years ago the Japanese village at Knightsbridge was one of the shows of London, there were some acrobats there who tossed on to the top surface of an umbrella a penny so that it alighted on its edge, and then, by turning round the stick of the umbrella rapidly, the coin was caused to rotate, but as the umbrella moved away underneath it the coin remained apparently stationary and standing upright, while by diminishing or increasing the angular velocity of the umbrella the penny was caused to run forwards or backwards. This is not a difficult trick to execute. The tendency of a body in motion to continue to move in a straight line is sometimes called itscentrifugal force. Thus, if a train is running roundacurve, ittendstomoveinastraightline, andisconstrainedonly by the pressure of the rails to move in the required direction. Hence it presses on the outer rail of the curve. This pressure can be diminished to some extent both by raising the outer rail, and by putting a guard rail, parallel and close to the inner rail, against which the wheels on that side also will press. An illustration of this fact occurred in a little known incident of the American civil war*. In the spring of 1862 a party of volunteers from the North made their way to the rear of the Southern armies and seized a train, intending to destroy, as they passed along it, the railway which was the main line of communication between various confederate corps and their base of operations. They were however detected and pursued. To save themselves, they stopped on a sharp curve and tore * Capturing a Locomotiveby W. Pittenger, London, 1882, p. 104.

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[PDF page 88] 72 MECHANICAL RECREATIONS. [CH. III up some rails so as to throw the engine which was following them off the line. Unluckily for themselves they were ignorant of dynamics and tore up the inner rails of the curve, an operation which did not incommode their pursuers. The second law gives us the means of measuring mass, force, and therefore work. A given agent in a given time can do only a definite amount of work. This is illustrated by the fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move any mass however large, yet what is gained in power is lost in speed—as the popular phrase runs. Montucla* inserted a striking illustration of this principle founded on the well-known story of Archimedes who is said to have declared to Hiero that, were he but given a fixed fulcrum, he could move the world. Montucla calculated the mass of the earth and, assuming that a man could work incessantly at the rate of116 foot-lbs. per second, which is a very high estimate, he found that it would take over three billion centuries,i.e. 3 × 1014 years, before a mass equal to that of the earth was moved as much as one inch against gravity at the surface of the earth: to move it one inch along a horizonal plane would take about 74000 centuries. Stability of Equilibrium. It is known to all those who have read the elements of mechanics that the centre of gravity of a body, which is resting in equilibrium under its own weight, must be vertically above its base: also, speaking generally, we may say that, if every small displace- ment has the effect of raising the centre of gravity, then the equilibrium is stable, that is, the body when left to itself will return to its original position; but, if a displacement has the effect of lowering the centre of gravity, then for that displacement the equilibrium is unstable; while, if every displacement does not alter the height above some fixed plane of the centre of gravity, then the equilibrium is neutral. In other words, if in order to cause a displacement work has to be done against the forces acting on the body, then for that displacement the equilibrium is stable, while if the forces do work the equilibrium is unstable. A good many of the simpler mechanical toys and tricks afford il- lustrations of this principle. * Ozanam, 1803 edition, vol. ii, p. 18; 1840 edition, p. 202.
