part V, pp. 89 — 141.

CH. IV] GEOMETRICAL RECREATIONS 81
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 length AB 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 end A is twisted through two right angles before it is gummed to B (the result of which is that the back of the strip at A is gummed over the front of the strip at B), then a cut along the line will produce only one ring. Next suppose that the end A is twisted once completely round {i.e. through four right angles) before it is gummed to B, 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 end A receives m half-twists, that is, is twisted through mir, 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 has m half-twists, and which are linked together \m times.
If m is odd we obtain a surface having only one side and one eJge. If this ring is cut along its mid-line, we obtain only one ring, but it has 2m half-twists, and if m is greater than unity it is knotted.
ADDENDUM.
Note. Page 64. One method of arranging 16 counters in 15 lines, as stated in the text, is as follows. Draw a regular re-entrant pentagon vertices A\j ^2) -^3? -^4> ^5,
* Vorstadien zur Topologic, Die Studieii, Gottingen, 1647, part x. B. R. 6
82 GEOMETRICAL RECREATIONS [CH. IV
points Biy ... B^. These latter points may be joined so as to form a smaller regular re-entrant pentagon whose sides intersect in five points Ci, ... Cs. The 16 points indicated are arranged as desired {The Canter- bury Puzzles, 1907, p. 140).
An arrangement of 18 counters in 9 rows, each containing 5 counters, can be obtained thus. From one angle, A of an equilateral triangle AA'A!\ draw lines AB, AE inside the triangle making any angles with A A'. Draw from A' and A" lines similarly placed in regard to A' A" and A" A. Let A'D' cut A"E" in F, and A'E' cut A"D" in G. Then AFG is a straight line. The 3 vertices of the triangle and the 15 points of inter- section of AD, AE, AF, with the similar pencils of Unes drawn from A'y A", will give an arrangement as required.
An arrangement of 19 counters in 10 rows, each containing 5 counters, can be obtained by placing counters at the 19 points of intersection of the 10 Hnes .^= ±a, x= ±6, y= ±a, y= ±b, y= ±x: of these points two are at infinity.
Note. Page 69. The Great Northern Shunting Problem is effected thus, (i) R pushes P into A. (ii) R returns, pushes ^ up to P in ^, couples Q to P, draws them both out to F, and then pushes them to E. (iii) P is now uncoupled, R takes Q back to A, and leaves it there, (iv) R returns to P, takes P back to (7, and leaves it there, (v) R running successively through F, i>, B comes to -4, draws Q out, and leaves it at B.
Note. Page 70. One solution of the Chifu-Chemulpo Puzzle is as follows. Move successively wagons 2, 3, 4 up, i.e. on to the loop line. [Then push 1 along the straight track close to 5 ; this is not a " move."] Next, move 4 down, i.e. on to the straight track and push it along to 1. Next, move 8 up, 3 down to the end of the track and keep it there tempo- rarily, 6 up, 2 down, e down, 3 up, 7 up. [Then push 5 to the end of the track and keep it there temporarily.] Next, move 7 down, 6 down, 2 up, 4 up. [Then push e along to 1.] Next, move 4 down to the end of the track and keep it there temporarily, 2 down, 5 up, 3 down, 6 up, 7 up, 8 down to the end of the track, e up, 5 down, 6 down, 7 down. In this solution we moved e down to the track at one end, then shifted it along the track, and finally moved it up to the loop from the other end of the track. We might equally well move e down to the track at one end, and finally move it back to the loop from the same ^nd. In this solution the pieces successively moved are 2, 3, 4, 4, e, 8, 7, 3, 2, 6, 5, 5, 6, 3, 2, 7, 2, 5, C, 3, 7, e, 8, 5, 6, 7.
83
CHAPTER y.
MECHANICAL RECREATIONS.
I proceed now to enumerate a few questions connected with mechanics which lead to results that seem to me interesting from a historical point of view or paradoxical. Problems in mechanics generally involve more difficulties than problems in arithmetic, algebra, or geometry, 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 I exclude all transcendental meclianics, and confine myself to questions which, like those treated in the preceding chapters, are of an elementary character. The results are well-known to mnthematicians.
I assume that the reader is acquainted with the funda- mental 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 iniiuence one another.
6-2
84j mechanical recreations [ch. V
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 (that is, 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 incon- ceivable, 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 the 1000 yards, the tortoise would still be 100 yards in front of him ; by the time he had covered these 100 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.
CH. V] MECHANICAL RECREATIONS 85
Zeno's Paradox on Time. The fallacy of Achilles and the Tortoise is usually explained by saying that though the time required to overtake 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 assump- tion that space and time are infinitely divisible, an assumption which he would not have admitted. 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, that t is the smallest conceivable interval, and suppose that three horizontal lines composed of three consecutive spans ahc, a'b'c', a"h"c" are placed so that a, a\ a" are vertically over one another, as also h, h\ h" and c, c', c". Imagine the second line moved as a whole one span to the right in the time r, and simultaneously the third line moved as a whole one span to the left. Then 6, a', c" will be vertically over one another. And in this duration r (which by hypothesis is indivisible) a' must have passed vertically over the space a"h" and the space 6V. Hence the duration is divisible, contrary to the hypothesis.
Tke Paradox of Tristram Shandy. Mr Russell has enun- ciated* 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 finish the work, however long he lived. But had he lived long enough, and not wearied of his task, then, even if his life had continued as eventfuUy as it began, no part of his biograpliy would have remained unwritten. For if he wrote the events of ihe first day in the first year, he would write the
* B. A. W. Russell, Principles oj Mathematics, Cambridge, 1903, vol. i, p. 368.
86 MECHANICAL RECREATIONS [CH. V
events of the nth day in the nth. 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 of space and time involve difficulties of a high order. Many of the resulting perplexities are due to the assumption that the number of things in a collection of them is greater than the number in a part of that collection. This is axiomatic for a finite number of things, but must not be assumed as being necessarily true of infinite collections.
Angular Motion. A non-mathematician finds additional difficulties in the idea of angular motion. For instance, there is a well-known proposition on motion in an equiangular spiral 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. To a non- mathematician the result seems paradoxical if not impossible.
The demonstration is as follows. The equiangular spiral is the trace of a point P, which moves along a line OP, the line OP turning round a fixed point 0 with uniform angular velocity while the distance of P from 0 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 con- volution immediately outside it. Inside any given convolu- tion 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 in a seconds, it will cover the next in ajn seconds, the next in ajn^ seconds, and so on, and will finally reach the pole in
(a -f- ajn ■\- a/n^ -\- ajn^ -\- )
seconds, that is, in anl{n — 1) seconds. The velocity is uniform,
CH. V] MECHANICAL RECREATIONS 87
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 diffi- culties suggested 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 from B to A a, passenger from A to B would meet and pass on his way, assuming that the journey either way takes 4J hours and that the trains start from each end every hour. The answer is 9. Or again, 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 is 2.
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 define force, but it is iu 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 statements are current may be fairly reckoned among mechanical paradoxes.
* The proposition is put in this form in J. Richard's Philosophic de$ Mathematiques, Paris, 1903, pp. 119—120.
88 MECHANICAL RECREATIONS [CH. V
The idea of force is difficult to grasp. How many people, for instance, could predict correctly what would happen in a question as simple as the following ? A rope (whose weight may be neglected) hangs over a smooth pulley ; it has one end fastened to a weight of 10 stone, and the other end to a sailor of weight 10 stone, the sailor and the weight hanging in the air. The sailor begins steadily to climb up the rope ; will the weight move at all ; and, if so, will it rise or fall ? In fact, it will rise.
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 not time to change their relative positions before there is sufficient space for them to drop vertically as if they were a solid body. On this principle depends the successful per- formance of numerous mechanical tricks and puzzles.
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 XX. 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,
CH. V] MECHANICAL RECREATIONS 89
though the full explanation of most of them also introduces other considerations. Thus it is a common feat to toss on to the top surface of an umbrella a penny so that it alights on its edge, and then, by turning round the stick of the umbrella rapidly, to cause the coin to rotate. By twisting the umbrella at the proper rate, the coin can be made to appear stationary and standing upright, though the umbrella is moving away underneath it, while by diminishing or increasing the angular velocity of the umbrella the penny can be made to run forwards or backwards. This is not a difficult trick to execute : it was introduced by Japanese conjurers.
The tendency of a body in motion to continue to move in a straight line is sometimes called its centrifugal force. Thus, if a train is running round a curve, it tends to move in a straight line, and is constrained only 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 inci- dent 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, intendins" 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 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, who were travelling at a high speed.
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 ♦ Capturing a Locomotive by W. Pittenger, London, 1882, p. 104.
90 MECHANICAL RECREATIONS [CH. V
fact that although, by means of a rigid lever and a fixed fulcrum, any force however small may be caused to move an}^ 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 inces- santly at the rate of 116 foot-lbs. per second, which is a very high estimate, he found that it would take over three billion centuries, i.e. 3 x 10^^ years, before a particle whose mass was 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 horizontal plane would take about 74,000 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 displacement 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 illustrations of this principle.
Magic Bottles\. Among the most common of such toys are the small bottles — trays of which may be seen any day in the streets of London — which keep always upright, and cannot
* Ozanam, 1803 edition, vol. ii, p. 18; 1840 edition, p. 202. t Ozanam, 1803 edition, vol. n, p. 15 ; 1840 edition, p. 201.
CH. V] MECHANICAL RECREATIOiNS 91
be upset until their owner orders them to lie down. Such a bottle is made of thin glass or varnished paper fixed to the plane surface of a solid hemisphere or smaller segment of a sphere. Now the distance of the centre of gravity of a homogeneous hemisphere from the centre of the sphere is three-eighths of the radius, and the mass of the glass or varnished paper is so small compared with the mass of the lead base that the centre of gravity of the whole bottle is still within the hemisphere. Let us denote the centre of the hemisphere by G, and the centre of gravity of the bottle by G.
If such a bottle is placed with the hemisphere resting on a horizontal plane and GO vertical, any small displacement on the plane will tend to raise G, and thus the equilibrium is stable. This may be seen also from the fact that when slightly dis- placed there is brought into play a couple, of whicli one force is the reaction of the table passing through C and acting vertically upward, and the other the weight of the bottle acting vertically downward at G. If G is below (7, this couple tends to restore the bottle to its original position.
If there is drop^^ed into the bottle a shot or nail so heavy as to raise the centre of gravity of the whole above C, then the equilibrium is unstable, and, if any small displacement is given, the bottle falls over on to its side.
Montucla says that in his time it was not uncommon to see boxes of tin soldiers mounted on lead hemispheres, and when the lid of the box was taken off the whole reg-iment sprang to attention.
In a similar way we may explain how to balance a pencil in a vertical position, with its point resting on the top of one's finger, an experiment which is described in nearly every book of puzzles*. This is effected by taking a penknife, of which one blade is opened through an angle of (say) 120°, and sticking the blade in the pencil so that the handle of the penknife is below the fiuger. The centre of gravity is thus brought below the point of support, and a small displacement given to the
■ Ex. gr. Oughtre>l, Muthematicall Rccrealions, p. 2-i ; Ozanam, 1803 eJilion, vol. II, p. 14; 1810 edition, p. 2U0.
92 MECHANICAL RECREATIONS [CH. V
pencil will raise the centre of gravity of the whole : thus the equilibrium is stable.
Other similar tricks are the suspension of a bucket over the edge of a table by a couple of sticks, and the balancing of a coin on the edge of a wine-glass by the aid of a couple of forks* — the sticks or forks being so placed that the centre of gravity of the whole is vertically below the point of support and its depth below it a maximum.
The toy representing a horseman, whose motion continually brings him over the edge of a table into a position which seems to ensure immediate destruction, is constructed in somewhat the same way. A wire has one end fixed to the feet of the rider; the wire is curved downwards and backwards, and at the other end is fixed a weight. When the horse is placed so that his hind legs are near the edge of the table and his fore- feet over the edge, the weight is under his hind feet. Thus the whole toy forms a pendulum with a curved instead of a straight rod. Hence the farther it swings over the table, the higher is the centre of gravity raised, and thus the toy tends to return to its original position of equilibrium.
An elegant modification of the prancing horse was brought out at Paris in 1890 in the shape of a toy made of tin and in the figure of a man-f-. The legs are pivoted so as to be movable about the thighs, but with a wire check to prevent too long a step, and the hands are fastened to the top of a fl -shaped wire weighted at its ends. If the figure is placed on a narrow sloping plauk or strip of wood passing between the legs of the n, then owing to the 0 -shaped wire any lateral displacement of the figure will raise its centre of gravity, and thus for any such displacement the equilibrium is stable. Hence, if a slight lateral disturbance is given, the figure will oscillate and will rest alternately on each foot: when it is supported by one foot the other foot under its own weight moves forwards, and thus the figure will walk down the plank though with a slight
* Oughtred, p. oO; Ozanam, 1803 edition, vol. ii, p. 12; 1840 edition, p. 199.
t La Nature, Paris, Much, 1891,
CH. V] MECHANICAL RECREATIONS 93
reeling motion. Shortly after the publication of the third edition of this book an improved form of this toy, in the shape of a walking elephant made in heavy metal, was issued in England, and probably in that form it is now familiar to all who are interested in noticing street toys.
Columbus's Egg. The toy known as Columbus's egg depends on the same principle as the magic bottle, though it leads to the converse result. The shell of the egg is made of tin and cannot be opened. Inside it and fastened to its base is a hollow truncated tin cone, and there is also a loose marble inside the shell. If the egg is held properly, the marble runs inside the cone and the egg will stand on its base, but so long as the marble is outside the cone, the egg cannot be made to stand on its base.
Cones running up hill*. The experiment to make a double cone run up hill depends on the same principle as the toys above described ; namely, on the tendency of a body to take a position so that its centre of gravity is as low as possible. In this case it produces the optical effect of a body moving by itself up a hill.
Usually the experiment is performed as follows. Arrange two sticks in the shape of a V, with the apex on a table and the two upper ends resting on the top edge of a book placed on the table. Take two equal cones fixed base to base, and place them with the curved surfaces resting on the sticks near the apex of the V, the common axis of the cones being horizontal and parallel to the edge of the book. Then, if properly arranged, the cones will run up the plane formed by the sticks.
The explanation is obvious. The centre of gravity of the cones moves in the vertical plane midway between the two sticks and it occupies a lower position as the points of contact on the sticks get farther apart. Hence as the cone rolls up the sticks its centre of gravity descends.
Perpetual Motion, The idea of making a machine which once set going would continue to go for ever by itself has been
* Ozunam, 18U3 edition, vol. ii, p. 49 ; 1840 editiou, p. 216.
94 MKCHANICAL RECREATIONS [CH. V
the ignis fatuus of self-taught mechanicians in much the same way as the quadrature of the circle has been that of self-taught geometricians.
Now the obvious meaning of the third law of motion is that a force is only one aspect of a stress, and that whenever a force is caused another equal and opposite one is brought also into existence — though it may act upon a different body, and thus be immaterial for the particular problem considered. The law however is capable of another interpretation*, namely, that the rate at which an agent does work (that is, its action) is equal to the rate at which work is done against it (that is, its reaction). If it is allowable to include in the reaction the rate at which kinetic energy is being produced, and if work is taken to include that done against molecular forces, then it follows from this interpretation that the work done by an agent on a system is equivalent to the total increase of energy, that is, the power of doing work. Hence in an isolated system the total amount of energy is constant. If this is granted, then since friction and some molecular dissipation of energy cannot be wholly prevented, it must be impossible to construct in an isolated system a machine capable of perpetual motion.
I do not propose to describe in detail the various machines for producing perpetual motion which have been suggested f, but the machine described below will serve to illustrate one of the assumptions commonly made by these inventors.
The machine to which I refer consists of two concentric vertical wheels in the same plane, and mounted on a horizontal axle through their centre, G. The space between the wheels is divided into compartments by spokes inclined at a constant angle to the radii to the points whence they are drawn, and each compartment contains a heavy bullet. This will be clear from the diagram. Apart from these bullets, the wheels would be in equilibrium. Each bullet tends to turn the wheels round their axle, and the moment which measures this tendency is
* Newton's Principles, last paragraph of the Scholium to the Laws of Motion, t Several of them have been described in H. Dirck's PerpeLaum Mobile, London, 1861, 2ud edition, 1870.
CH. V]
MECHANICAL llECIlEATlO^^iS
95
the product of the weight of the bullet and its distance from the vertical through G.
The idea of the constructors of such machines was that, as the bullet in any compartment would roll under gravity to the lowest point of the compartment, the bullets on the right-hand side of the wheel in the diagram would be farther from the vertical throuo^h C than those on the left. Hence the sum of
the moments of the weights of the bullets on the right would be greater than the sum of the moments of those on the left. Thus the wheels would turn continually in the same direction as the hands of a watch. The fallacy in the argument is obvious.
Another large group of machines for producing perpetual motion depended on the use of a magnet to raise a mass which was then allowed to fall under gravity. Thus, if the bob of a simple pendulum was made of iron, it was thought that magnets fixed near the highest points which were reached by the bob in the swing of the pendulum would draw the bob up to the same height in each swing and thus give perpetual motion, but the inventors omitted to notice that the bob of the pendulum would gradually get magnetised.
Of course it is only in isolated systems that the total amount of energy is constant, and, if a source of external energy can be obtained from which energy is continually introduced into the system, perpetual motion is, in a sense, possible ; though even here materials would ultimately wear out. Streams, wind, the
96 MECHANICAL RECREATIONS [CH. V
solar heat, and the tides are among the more obvious of such sources.
There was at Paris in the latter half of the eighteenth century a clock which was an ingenious illustration of such perpetual motion*. The energy which was stored up in it to maintain the motion of the pendulum was provided by the expansion of a silver rod. This expansion was caused by the daily rise of temperature, and by means of a train of levers it wound up the clock. There was a disconnecting apparatus, so that the contraction due to a fall of temperature produced no effect, and there was a similar arrangement to prevent over- winding. I believe that a rise of eight or nine degrees Fahrenheit was sufficient to wind up the clock for twenty-four hours.
By utilizing the rise and fall of the barometer, James Cox, a London jeweller of the eighteenth century, produced, in an analogous way, a clock f which ran continuously without winding up.
I have in my possession a watch which produces the same effect by somewhat different means. Inside the case is a steel weight, and if the watch is carried in a pocket this weight rises and falls at every step one takes, somewhat after the manner of a pedometer. The weight is raised by the action of the person who has it in his pocket in taking a step, and in falling it winds up the spring of the watch. On the face is a small dial showing the number of hours for which the watch is wound up. As soon as the hand of this dial points to fifty-six hours, the train of levers which winds up the watch disconnects automatically, so as to prevent over- winding the spring, and it reconnects again as soon as the watch has run down eight hours. The watch is an excellent time-keeper, and a walk of about a couple of miles is sufficient to wind it up for twenty- four hours.
♦ Ozanam, 1S03 edition, vol. ii, p. 105 ; 1840 edition, p. 238. t A full description of the mechanism will be found in the English Mechanic, April 30, 1909, pp. 288—289.
CH. V] MECHANICAL RECREATIONS 97
Models. I may add here the observation, which is well known to mathematicians, but is a perpetual source of disap- pointment to ignorant inventors, that it frequently happens that an accurate model of a machine will work satisfactoril}^ while the machine itself will not do so.
One reason for this is as follows. If all the parts of a model are magnified in the same proportion, say m, and if thereby a line in it is increased in the ratio m : 1, then the areas and volumes in it will be increased respectively in the ratios m? : 1 and m^ : 1. For example, if the side of a cube is doubled then a face of it will be increased in the ratio 4 : 1 and its volume will be increased in the ratio 8:1.
Now if all the linear dimensions are increased m times then some of the forces that act on a machine (such, for example, as the weight of part of it) will be increased I'n? times, while others which depend on area (such as the sustaining power of a beam) will be increased only m? times. Hence the forces that act on the machine and are brought into play by the various parts may be altered in different proportions, and thus the machine may be incapable of producing results similar to those which can be produced by the model.
The same argument has been adduced in the case of animal life to explain why very large specimens of any particular breed or species are usually weak. For example, if the linear dimen- sions of a bird were increased n times, the work necessary to give the power of flight would have to be increased no less than n^ times*. Again, if the linear dimensions of a man of height 5 ft. 10 in. were increased by one-seventh his height would become 6 ft. 8 in., but his weight would be increased in the ratio 512 : 843 {i.e. about half as much again), while the cross sections of his legs, which would have to bear this weight, would be increased only in the latio 64:49; thus in some respects he would be less etficieut than before. Of course the increased dimensions, length of limb, or size of muscle might be of greater advantage than the relative loss of strength ; hence the problem of what are the most efficient
* Helmholtz, Gesammelte Ahhandlunrjen, Leipzig, 1881, vol. i, p. 165. B. R. 7
98 MECHANICAL RECREATIOXS [CH. V
proportions is not simple, but the above argument will serve to illustrate the fact that the working of a machine may not be similar to that of a model of it.
Leaving now these elementary considerations I pass on to some other mechanical questions.
Sailing quicker than the Wind. As a kinematical paradox I may allude to the possibility of sailing quicker than the wind hloivs, a fact which strikes many people as curious.
The explanation* depends on the consideration of the velocity of the wind relative to the boat. Perhaps, however, a non-mathematician will find the solution simplified if I con- sider first the effect of the wind-pressure on the back of the sail which drives the boat forward, and second the resistance to motion caused by the sail being forced through the air.
When the wind is blowing against a plane sail the resultant pressure of the wind on the sail may be resolved into two components, one perpendicular to the sail (but which in general is not a function only of the component velocity in that direc- tion, though it vanishes when that component vanishes) and the other parallel to its plane. The latter of these has no effect on the motion of the ship. The component perpen- dicular to the sail tends to move the ship in that direction. This pressure, normal to the sail, may be resolved again into two components, one in the direction of the keel of the boat, the other in the direction of the beam of the boat. The former component drives the boat forward, the latter to lee- ward. It is the object of a boat-builder to construct the boat on lines so that the resistance of the water to motion forward shall be as small as possible, and the resistance to motion in a perpendicular direction (i.e. to leeward) shall be as large as possible ; and I will assume for the moment that the former of these resistances may be neglected, and that the latter is so large as to render motion in that direction impossible.
Now, as the boat moves forward, the pressure of the air on the front of the sail will tend to stop the motion. As
* Ozanam, 1803 edition, vol. iii, pp. 359, 367 ; 1840 edition, pp. 540, 543.