The Conservation of Energy/Chapter 2

by Balfour Stewart

CHAPTER II.

MECHANICAL ENERGY AND ITS CHANGE INTO HEAT.

Energy of Position. A Stone high up.

34. In the last chapter it was shown what is meant by energy, and how it depends upon the velocity of a moving body; and now let us state that this same energy or power of doing work may nevertheless be possessed by a body absolutely at rest. It will be remembered (Art. 26) that in one case where a kilogramme was shot vertically upwards, we supposed it to be caught at the summit of its flight and lodged on the top of a house. Here, then, it rests without motion, but yet not without the power of doing work, and hence not without energy. For we know very well that if we let it fall it will strike the ground with as much velocity, and, therefore, with as much energy, as it had when it was originally projected upwards. Or we may, if we choose, make use of its energy to assist us in driving in a pile, or utilize it in a multitude of ways.

In its lofty position it is, therefore, not without energy, but this is of a quiet nature, and not due in the least to motion. To hat, then, is it due? We reply—to the poisition which the kilogramme occupies at the top of the house. For just as a body in motion is a very different thing (as regards energy) from a body at rest, so is a body at the top of a house a very different thing from a body at the bottom.

To illustrate this, we may suppose that two men of equal activity and strength are fighting together, each having his pile of stones with which he is about to belabour his adversary. One man, however, has secured for himself and his pile an elevated position on the top of a house, while his enemy has to remain content with a position at the bottom. Now, under these circumstances, you can at once tell which of the two will gain the day—evidently the man on the top of the house, and yet not on account of his own superior energy, but rather on account of the energy which he derives from the elevated position of his pile of stones. We thus see that there is a kind of energy derived from position, as well as a kind derived from velocity, and we shall, in future, call the former energy of position, and the latter energy of motion.

A Head of Water.

35. In order to vary our illustration, let us suppose there are two mills, one with a large pond of water near it and at a high level, while the other has also a pond, but at a lower level than itself We need hardly ask which of the two is likely to work—clearly the one with the pond at a low level can derive from it no advantage whatever, while the other may use the high level pond, or head of water, as this is sometimes called, to drive its wheel, and do its work. There is, thus, a great deal of work to be got out of water high up—real substantial work, such as grinding corn or thrashing it, or turning wood or sawing it. On the other hand, there is no work at all to be got from a pond of water that is low down.

A Cross-bow bent. A Watch wound up.

36. In both of the illustrations now given, we have used the force of gravity as that force against which we are to do work, and in virtue of which a stone high up, or a head of water, is in a position of advantage, and has the power of doing work as it falls to a lower level. But there are other forces besides gravity, and, with respect to these, bodies may be in a position of advantage and be able to do work just as truly as the stone, or the head of water, in the case before mentioned.

Let us take, for instance, the force of elasticity, and consider what happens in a cross-bow. When this is bent, the bolt is evidently in a position of advantage with regard to the elastic force of the bow; and when it is discharged, this energy of position of the bolt is converted into energy of motion, just as, when a stone on the top of a house is allowed to fall, its energy of position is converted into that of actual motion.

In like manner a watch wound up is in a position of advantage with respect to the elastic force of the mainspring, and as the wheels of the watch move this is gradually converted into energy of motion.

Advantage of Position.

37. It is, in fact, the fate of all kinds of energy of position to be ultimately converted into energy of motion.

The former may he compared to money in a bank, or capital, the latter to money which we are in the act of spending; and just as, when we have money in a bank, we can draw it out whenever we want it, so, in the case of energy of position, we can make use of it whenever we please. To see this more clearly, let us compare together a watermill driven by a head of water, and a windmill driven by the wind. In the one case we may turn on the water whenever it is most convenient for us, but in the other we must wait until the wind happens to blow. The former has all the independence of a rich man; the latter, all the obsequiousness of a poor one. If we pursue the analogy a step further, we shall see that the great capitalist, or the man who has acquired a lofty position, is respected because he has the disposal of a great quantity of energy; and that whether he be a nobleman or a sovereign, or a general in command, he is powerful only from having something which enables him to make use of the services of others. When the man of wealth pays a labouring man to work for him, he is in truth converting so much of his energy of position into actual energy, just as a miller lets out a portion of his head of water in order to do some work by its means.

Transmutations of Visible Energy.—A Kilogramme shot upwards.

38. We have thus endeavoured to show that there is an energy of repose as well as a living energy, an energy of position as well as of motion; and now let us trace the changes which take place in the energy of a weight, shot vertically upwards, as it continues to rise. It starts with a certain amount of energy of motion, but as it ascends, this is by degrees changed into that of position, until, when it gets to the top of its flight, its energy is entirely due to position.

To take an example, let us suppose that a kilogramme is projected vertically upwards with the velocity of 19.6 metres in one second. According to the formula of Art. 28, it contains 19.6 units of energy due to its actual velocity.

If we examine it at the end of one second, we shall find that it has risen 14.7 metres in height, and has now the velocity of 9.8. This velocity we know (Art. 26) denotes an amount of actual energy equal to 4.9, while the height reached corresponds to an energy of position equal to 14.7. The kilogramme has, therefore, at this moment a total energy of 19.6, of which l4.7 units are due to position, and 4.9 to actual motion.

If we next examine it at the end of another second, we shall find that it has just been brought to rest, so that its energy of motion is nil; nevertheless, it has succeeded in raising itself 19.6 metres in height, so that its energy of position is 19.6.

There is, therefore, no disappearance of energy during the rise of the kilogramme, but merely a gradual change from one kind to another. It starts with actual energy, and this is gradually changed into that of position; but if, at any stage of its ascent, we add together the actual energy of the kilogramme, and that due to its position, we shall find that their sum always remains the same.

39. Precisely the reverse takes place when the kilogramme begins its descent. It starts on its downward journey with no energy of motion whatever, but with a certain amount of energy of position; as it falls, its energy of position becomes less, and its actual energy greater, the sum of the two remaining constant throughout, until, when it is about to strike the ground, its energy of position has been entirely changed into that of actual motion, and it now approaches the ground with the velocity, and, therefore, with the energy, which it had when it was originally projected upwards.

The Inclined Plane.

40. We have thus traced the transmutations, as regards energy, of a kilogramme shot vertically upwards, and allowed to fall again to the earth, and we may now vary our hypothesis by making the kilogramme rise vertically, but descend by means of a smooth inclined plane without friction—imagine in fact, the kilogramme to be shaped like a ball or roller, and the plane to be perfectly smooth. Now, it is well known to all students of dynamics, that in such a case the velocity which the kilogramme has when it has reached the bottom of the plane will be equal to that which it would have had if it had been dropped down vertically through the same height, and thus, by introducing a smooth inclined plane of this kind, you neither gain nor lose anything as regards energy.

In the first place, you do not gain, for think what would happen if the kilogramme, when it reached the bottom of the inclined plane, should have a greater velocity than you gave it originally, when you shot it up. It would evidently be a profitable thing to shoot up the kilogramme vertically, and bring it down by means of the plane, for you would get back more energy than you originally spent upon it, and in every sense you would be a gamer. You might, in fact, by means of appropriate apparatus, convert the arrangement into a perpetual motion machine, and go on accumulating energy without limit—but this is not possible.

On the other hand, the inclined plane, unless it be rough and angular, will not rob you of any of the energy of the kilogramme, but will restore to you the full amount, when once the bottom has been reached. Nor does it matter what be the length or shape of the plane, or whether it be straight, or curved, or spiral, for in all cases, if it only be smooth and of the same vertical height, you will get the same amount of energy by causing the kilogramme to fall from the top to the bottom.

41. But while the energy remains the same, the time of descent will vary according to the length and shape of the plane, for evidently the kilogramme will take a longer time to descend a very sloping plane than a very steep one. In fact, the sloping plane will take longer to generate the requisite velocity than the steep one, but both will have produced the same result as regards energy, when once the kilogramme has arrived at the bottom.

Functions of a Machine.

42. Our readers are now beginning to perceive that energy cannot be created, and that by no means can we coax or cozen Dame Nature into giving us back more than we are entitled to get. To impress this fundamental principle still more strongly upon our minds, let us consider in detail one or two mechanical contrivances, and see what they amount to as regards energy.
Fig.1.

Let us begin with the second system of pulleys. Here we have a power P attached to the one end of a thread, which passes over all the pulleys, and is ultimately attached, by its other extremity, to a hook in the upper or fixed block. The weight W is, on the other hand, attached to the lower or moveable block, and rises with it. Let us suppose that the pulleys are without weight and the cords without friction, and that W is supported by six cords, as in the figure. Now, when there is equilibrium in this machine, it is well known that W will be equal to six times P; that is to say, a power of one kilogramme will, in such a machine, balance or support a weight of six kilogrammes. If P be increased a single grain more, it will overbalance W, and P will descend, while W will begin to rise. In such a case, after P has descended, say six metres, its weight being, say, one kilogramme, it has lost a quantity of energy of position equal to six units, since it is at a lower level by six metres than it was before. We have, in fact, expended upon our machine six units of energy. Now, what return have we received for this expenditure? Our return is clearly the rise of W, and mechanicians will tell us that in this case W will have risen one metre.

But the weight of W is six kilogrammes, and this having been raised one metre represents an energy of position equal to six. We have thus spent upon our machine, in the fall of P, an amount of energy equal to six units, and obtained in the rise of W an equivalent amount equal to six units also. We have, in truth, neither gained nor lost energy, but simply changed it into a form more convenient for our use.

Fig. 2. 43. To impress this truth still more strongly, let us take quite a different machine, such as the hydrostatic press. Its mode of action will be perceived from Fig. 2. Here we have two cylinders, a wide and a narrow one, which are connected together at the bottom by means of a strong tube. Each of these cylinders is provided with a water-tight piston, the space beneath being filled with water. It is therefore manifest, since the two cylinders are connected together, and since water is incompressible, that when we push down the one piston the other will be pushed up. Let us suppose that the area of the small piston is one square centimetre,^[1] and that of the large piston one hundred square centimetres, and let us apply a weight of ten kilogrammes to the smaller piston. Now, it is known, from the laws of hydrostatics, that every square centimetre of the larger piston will be pressed upwards with the force of ten kilogrammes, so that the piston will altogether mount with the force of 1000 kilogrammes—that is to say, it will raise a weight of this amount as it ascends.

Here, then, we have a machine in virtue of which a pressure of ten kilogrammes on the small piston enables the large piston to rise with the force of 1000 kilogrammes. But it is very easy to see that, while the small piston falls one metre, the large one will only rise one centimetre. For the quantity of water under the pistons being always the same, if this be pushed down one metre in the narrow cylinder, it will only rise one centimetre in the wide one.

Let us now consider what we gain by this machine. The power of ten kilogrammes applied to the smaller piston is made to fall through one metre, and this represents the amount of energy which we have expended upon our machine, while, as a return, we obtain 1000 kilogrammes raised through one single centimetre. Here, then, as in the case of the pulleys, the return of energy is precisely the same as the expenditure, and, provided we ignore friction, we neither gain nor lose anything by the machine. All that we do is to transmute the energy into a more convenient form—what we gain in power we lose in space; but we are willing to sacrifice space or quickness of motion in order to obtain the tremendous pressure or force which we get by means of the hydrostatic press.

Principle of Virtual Velocities.

44. These illustrations will have prepared our readers to perceive the true function of a machine. This was first clearly defined by Galileo, who saw that in any machine, no matter of what kind, if we raise a large weight by means of a small one, it will be found that the small weight, multiplied into the space through which it is lowered, will exactly equal the large weight, multiplied into that through which it is raised.

This principle, known as that of virtual velocities, enables us to perceive at once our true position. We see that the world of mechanism is not a manufactory, in which energy is created, but rather a mart, into which we may bring energy of one kind and change or barter it for an equivalent of another kind, that suits us better—but if we come with nothing in our hand, with nothing we shall most assuredly return. A machine, in truth, does not create, but only transmutes, and this principle will enable us to tell, without further knowledge of mechanics, what are the conditions of equilibrium of any arrangement.

For instance, let it be required to find those of a lever, of which the one arm is three times as long as the other. Here it is evident that if we overbalance the lever by a single grain, so as to cause the long arm with its power to fall down while the short one with its weight rises up, then the long arm will fall three inches for every inch through which the short arm rises; and hence, to make up for this, a single kilogramme on the long arm will balance three kilogrammes on the short one, or the power will be to the weight as one is to three.
Fig. 3.

45. Or, again, let us take the inclined plane as represented in Fig. 3. Here we have a smooth plane and a weight held upon it by means of a power P, as in the figure. Now, if we overbalance P by a single grain, we shall bring the weight W from the bottom to the top of the plane. But when this has taken place, it is evident that P has fallen through a vertical distance equal to the length of the plane, while on the other hand W has only risen through a vertical distance equal to the height. Hence, in order that the principle of virtual velocities shall hold, we must have P multiplied into its fall equal to W multiplied into its rise, that is to say,

$P\times$ Length of plane $=W\times$ Height of plane,

or ${\frac {P}{W}}={\frac {Height.}{Length.}}$

What Friction does.

46. The two examples now given are quite sufficient to enable our readers to see the true function of a machine, and they are now doubtless disposed to acknowledge that no machine will give back more energy than is spent upon it. It is not, however, equally clear that it will not give back less; indeed, it is a well-known fact that it constantly does so. For we have supposed our machine to be without friction—but no machine is without friction—and the consequence is that the available out-come of the machine is more or less diminished by this drawback. Now, unless we are able to see clearly what part friction really plays, we cannot prove the conservation of energy. We see clearly enough that energy cannot be created, but we are not equally sure that it cannot be destroyed; indeed, we may say we have apparent grounds for believing that it is destroyed—that is our present position. Now, if the theory of the conservation of energy be true—that is to say, if energy is in any sense indestructible—friction will prove itself to be, not the destroyer of energy, but merely the converter of it into some less apparent and perhaps less useful form.

47. We must, therefore, prepare ourselves to study what friction really does, and also to recognize energy in a form remote from that possessed by a body in visible motion, or by a head of water. To friction we may add percussion, as a process by which energy is apparently destroyed; and as we have (Art. 39) considered the case of a kilogramme shot vertically upwards, demonstrating that it will ultimately reach the ground with an energy equal to that with which it was shot upwards, we may pursue the experiment one step further, and ask what becomes of its energy after it has struck the ground and come to rest? We may vary the question by asking what becomes of the energy of the smith's blow after his hammer has struck the anvil, or what of the energy of the cannon ball after it has struck the target, or what of that of the railway train after it has been stopped by friction at the break-wheel? All these are cases in which percussion or friction appears at first sight to have destroyed visible energy; but before pronouncing upon this seeming destruction, it clearly behoves us to ask if anything else makes its appearance at the moment when the visible energy is apparently destroyed. For. after all, energy may be like the Eastern magicians, of whom we read that they had the power of changing themselves into a variety of forms, but were nevertheless very careful not to disappear altogether.

When Motion is destroyed, Heat appears.

48. Now, in reply to the question we have put, it may be confidently asserted that whenever visible energy is apparently destroyed by percussion or friction, something else makes its appearance, and that something is heat. Thus, a piece of lead placed upon an anvil may be greatly heated by successive blows of a blacksmith's hammer. The collision of flint and steel will produce heat, and a rapidly-moving cannon ball, when striking against an iron target, may even be heated to redness. Again, with regard to friction, we know that on a dark night sparks are seen to issue from the break-wheel which is stopping a railway train, and we know, also, that the axles of railway carriages get alarmingly hot, if they are not well supplied with grease.

Finally, the schoolboy will tell us that he is in the habit of rubbing a brass button upon the desk, and applying it to the back of his neighbour's hand, and that when his own hand has been treated in this way, he has found the button unmistakeably hot.

Heat a species of Motion.

49. For a long time this appearance of heat by friction or percussion was regarded as inexplicable, because it was believed that heat was a kind of matter, and it was difficult to understand where all this heat came from. The partisans of the material hypothesis, no doubt, ventured to suggest that in such processes heat might be drawn from the neighbouring bodies, so that the Caloric (which was the name given to the imaginary substance of heat) was squeezed or rubbed out of them, according as the process was percussion or friction. But this was regarded by many as no explanation, even before Sir Humphry Davy, about the end of last century, clearly showed it to be untenable.

50. Davy's experiments consisted in rubbing together two pieces of ice until it was found that both were nearly melted, and he varied the conditions of his experiments in such a manner as to show that the heat produced in this case could not be abstracted from the neighbouring bodies.

51. Let us pause to consider the alternatives to which we are driven by this experiment. If we still choose to regard heat as a substance, since this has not teen taken from the surrounding bodies, it must necessarily have been created in the process of friction. But if we choose to regard heat as a species of motion, we have a simpler alternative, for, inasmuch as the energy of visible motion has disappeared in the process of friction, we may suppose that it has been transformed into a species of molecular motion, which we call heat; and this was the conclusion to which Davy came.

52. About the same time another philosopher was occupied with a similar experiment Count Rumford was superintending the boring of cannon at the arsenal at Munich, and was forcibly struck with the very great amount of heat caused by this process. The source of this heat appeared to him to be absolutely inexhaustible, and, being unwilling to regard it as the creation of a species of matter, he was led like Davy to attribute it to motion.

53. Assuming, therefore, that heat is a species of motion, the next point is to endeavour to comprehend what kind of motion it is, and in what respects it is different from ordinary visible motion. To do this, let us imagine a railway carriage, full of passengers, to be whirling along at a great speed, its occupants quietly at ease, because, although they are in rapid motion, they are all moving at the same rate and in the same direction. Now, suppose that the train meets with a sudden check;—a disaster is the consequence, and the quiet placidity of the occupants of the carriage is instantly at an end.

Even if we suppose that the carriage is not broken up and its occupants killed, yet they are all in a violent state of excitement; those fronting the engine are driven with force against their opposite neighbours, and are, no doubt, as forcibly repelled, each one taking care of himself in the general scramble. Now, we have only to substitute particles for persons, in order to obtain an idea of what takes place when percussion is converted into heat. We have, or suppose we have, in this act the same violent collision of atoms, the same thrusting forward of A upon B, and the same violence in pushing back on the part of B—the same struggle, confusion, and excitement—the only difference being that particles are heated instead of human beings, or their tempers.

54. We are bound to acknowledge that the proof which we have now given is not a direct one; indeed, we have, in our first chapter, explained the impossibility of our ever seeing these individual particles, or watching their movements; and hence our proof of the assertion that heat consists in such movements cannot possibly be direct. We cannot see that it does so consist, but yet we may feel sure, as reasonable beings, that we are right in our conjecture.

In the argument now given, we have only two alternatives to start with—either heat must consist of a motion of particles or, when percussion or friction is converted into heat, a peculiar substance called caloric must be created, for if heat be not a species of motion it must necessarily be a species of matter. Now, we have preferred to consider heat, as a species of motion to the alternative of supposing the creation of a peculiar kind of matter.

55. Nevertheless, it is desirable to have something to say to an opponent who, rather than acknowledge heat to be a species of motion, will allow the creation of matter. To such an one we would say that innumerable experiments render it certain that a hot body is not sensibly heavier than a cold one, so that if heat be a species of matter it is one that is not subject to the law of gravity. If we burn iron wire in oxygen gas, we are entitled to say that the iron combines with the oxygen, because we know that the product is heavier than the original iron by the very amount which the gas has lost in weight. But there is no such proof that during combustion the iron has combined with a substance called caloric, and the absence of any such proof is enough to entitle us to consider heat to be a species of motion, rather than a species of matter.

Heat a Backward and Forward Motion.

56. We shall now suppose that our readers have assented to our proposition that heat is a species of motion. It is almost unnecessary to add that it must be a species of backward and forward motion; for nothing is more clear than that a heated substance is not in motion as a whole, and will not, if put upon a table, push its way from the one end to the other.

Mathematicians express this peculiarity by saying that, although there is violent internal motion among the particles, yet the centre of gravity of the substance remains at rest; and since, for most purposes, we may suppose a body to act as if concentrated at its centre of gravity, we may say that the body is at rest.

57. Let us here, before proceeding further, borrow an illustration from that branch of physics which treats of sound. Suppose, for instance, that a man is accurately balanced in a scale-pan, and that some water enters his ear; of course he will become heavier in consequence, and if the balance be sufficiently delicate, it will exhibit the difference. But suppose a sound or a noise enters his car, he may say with truth that something has entered, but yet that something is not matter, nor will he become one whit heavier in consequence of its entrance, and he will remain balanced as before. Now, a man into whose ear sound has entered may be compared to a substance into which heat has entered; we may therefore suppose a heated body to be similar in many respects to a sounding-body, and just as the particles of a sounding body move backwards and forwards, so we may suppose that the particles of a heated body do the same.

We shall take another opportunity (Art. 162) to enlarge upon this likeness; but, meanwhile, we shall suppose that our readers perceive the analogy.

Mechanical Equivalent of Heat.

58. We have thus come to the conclusion that when any heavy body, say a kilogramme weight, strikes the ground, the visible energy of the kilogramme is changed into heat; and now, having established the fact of a relationship between these two forms of energy, our next point is to ascertain according to what law the heating effect depends upon the height of fall. Let us, for instance, suppose that a kilogramme of water is allowed to drop from the height of 848 metres, and that we have the means of confining to its own. particles and retaining there the heating effect produced. Now, we may suppose that its descent is accomplished in two stages; that, first of all, it falls upon a platform from the height of 424 metres, and gets heated in consequence, and that then the heated mass is allowed to fall other 424 metres. It is clear that the water will now be doubly heated; or, in other words, the heating effect in such a case will be proportional to the height through which the body falls—that is to say, it will be proportional to the actual energy which the body possesses before the blow has changed this into heat. In fact, just as the actual energy represented by a fall from a height is proportional to the height, so is the heating effect, or molecular energy, into which the actual energy is changed proportional to the height also. Having established this point, we now wish to know through how many metres a kilogramme of water must fall in order to be heated one degree centigrade.

59. For a precise determination of this important point, we are indebted to Dr. Joule, of Manchester, who has, perhaps, done more than any one else to put the science of energy upon a sure foundation. Dr. Joule made numerous experiments, with the view of arriving at the exact relation between mechanical energy and heat; that is to say, of determining the mechanical equivalent of heat. In some of the most important of these he took advantage of the friction of fluids.

60. These experiments were conducted in the following manner. A certain fixed weight was attached to a pulley, as in the figure. The weight had, of course, a tendency

Fig. 4. to descend, and hence to turn the pulley round. The pulley had its axle supported upon friction wheels, at ƒ and ƒ, by means of which the friction caused by the movement ot the pulley was very much reduced. A string, passing over the circumference of the pulley, was wrapped round r, so that, as the weight descended, the pulley moved round, and the string of the pulley caused r to rotate very rapidly. Now, the motion of the axis r was conducted within the covered box B, where there was attached to r a system of paddles, of which a sketch is given in figure; and therefore, as r moved, these paddles moved also. There were, altogether, eight sets of these paddles revolving between four stationary vanes. If, therefore, the box were full of liquid, the paddles and the vanes together would churn it about, for these stationary vanes would prevent the liquid being carried along by the paddles in the direction of rotation.

Now, in this experiment, the weight was made to descend through a certain fixed distance, which was accurately measured. As it descended, the paddles were set in motion, and the energy of the descending weight was thus made to churn, and hence to heat some water contained in the box B. When the weight had descended a certain distance, by undoing a small peg p, it could be wound up again without moving the paddles in B, and thus the heating effect of several falls of the weight could be accumulated until this became so great as to be capable of being accurately measured by a thermometer. It ought to be mentioned that great care was taken in these experiments, not only to reduce the friction of the axles of the pulley as much as possible, but also to estimate and correct for this friction as accurately as possible; in fact, every precaution was taken to make the experiment successful.

61. Other experiments were made by Joule, in some of which a disc was made to rotate against another disc of cast-iron pressed against it, the whole arrangement being immersed in a cast-iron vessel filled with mercury. From all these experiments, Dr. Joule concluded that the quantity of heat produced by friction, if we can preserve and accurately measure it, will always be found proportional to the quantity of work expended He expressed this proportion by stating the number of units of work in kilogrammetres necessary to raise by 1° C. the temperature of one kilogramme of water. This was 424, as determined by his last and most complete experiments; and hence we may conclude that if a kilogramme of water be allowed to fall through 424 metres, and if its motion be then suddenly stopped, sufficient heat will be generated to raise the temperature of the water through 1° C, and so on, in the same proportion.

62. Now, if we take the kilogrammetre as our unit of work, and the heat necessary to raise a kilogramme of water 1° C. as our unit of heat, this proportion may be expressed by saying that one heat unit is equal to 424 units of work.

This number is frequently spoken of as the mechanical equivalent of heat; and in scientific treatises it is denoted by J., the initial of Dr. Joule's name.

63. We have now stated the exact relationship that subsists between mechanical energy and heat, and before proceeding further with proofs of the great law of conservation, we shall endeavour to make our readers acquainted with other varieties of energy, on the ground that it is necessary to penetrate the various disguises that our magician assumes before we can pretend to explain the principles that actuate him in his transformations.

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1. That is to say, a square the side of which is one centimetre, or the hundredth part of a metre.