(222.) A MACHINE is an instrument by which force or motion may be transmitted and modified as to its quantity and direction. There are two ways in which a machine may be applied, and which give rise to a division of mechanical science into parts denominated STATICS and DYNAMICS; the one including the theory of equilibrium, and the other the theory of motion. When a machine is considered statically, it is viewed as an instrument by which forces of determinate quantities and direction are made to balance other forces of other quantities and other directions. If it be viewed dynamically, it is considered as a means by which certain motions of determinate quantity and direction may be made to produce other motions in other directions and quantities. It will not be convenient, however, in the present treatise, to follow this division of the subject. We shall, on the other hand, as hitherto, consider the phenomena of equilibrium and motion together.

The effects of machinery are too frequently described in such a manner as to invest them with the appearance of paradox, and to excite astonishment at what appears to contradict the results of the most common experience. It will be our object here to take a different course, and to attempt to show that those effects which have been held up as matters of astonishment are the necessary, natural, and obvious results of causes adapted to produce them in a manner analogous to the objects of most familiar experience.

(223.) In the application of a machine there are three things to be considered. 1. The force or resistance which is required to be sustained, opposed, or overcome. 2. The force which is used to sustain, support, or overcome that resistance. 3. The machine itself by which the effect of this latter force is transmitted to the former. Of whatever nature be the force or the resistance which is to be sustained or overcome, it is technically called the weight, since, whatever it be, a weight of equivalent effect may always be found. The force which is employed to sustain or overcome it is technically called the power.

(224.) In expressing the effect of machinery it is usual to say that the power sustains the weight; but this, in fact, is not the case, and hence arises that appearance of paradox which has already been alluded to. If, for example, it is said that a power of one ounce sustains the weight of one ton, astonishment is not unnaturally excited, because the fact, as thus stated, if the terms be literally interpreted, is physically impossible. No power less than a ton can, in the ordinary acceptation of the word, support the weight of a ton. It will, however, be asked how it happens that a machine appears to do this? how it happens that by holding a silken thread, which an ounce weight would snap, many hundred weight may be sustained? To explain this it will only be necessary to consider the effect of a machine, when the power and weight are in equilibrium.

(225.) In every machine there are some fixed points or props; and the arrangement of the parts is always such, that the pressure, excited by the power or weight, or both, is distributed among these props. If the weight amount to twenty hundred, it is possible so to distribute it, that any proportion, however great, of it may be thrown on the fixed points or props of the machine; the remaining part only can properly be said to be supported by the power, and this part can never be greater than the power. Considering the effect in this way, it appears that the power supports just so much of the weight and no more as is equal to its own force, and that all the remaining part of the weight is sustained by the machine. The force of these observations will be more apparent when the nature and properties of the mechanic powers and other machines have been explained.

(226.) When a machine is considered dynamically, its effects are explained on different principles. It is true that, in this case, a very small power may elevate a very great weight; but nevertheless, in so doing, whatever be the machine used, the total expenditure of power, in raising the weight through any height, is never less than that which would be expended if the power were immediately applied to the weight without the intervention of any machine. This circumstance arises from an universal property of machines by which the velocity of the weight is always less than that of the power, in exactly the same proportion as the power itself is less than the weight; so that when a certain power is applied to elevate a weight, the rate at which the elevation is effected is always slow in the same proportion as the weight is great. From a due consideration of this remarkable law, it will easily be understood, that a machine can never diminish the total expenditure of power necessary to raise any weight or to overcome any resistance. In such cases, all that a machine ever does or ever can do, is to enable the power to be expended at a slow rate, and in a more advantageous direction than if it were immediately applied to the weight or the resistance.

Let us suppose that P is a power amounting to an ounce, and that W is a weight amounting to 50 ounces, and that P elevates W by means of a machine. In virtue of the property already stated, it follows, that while P moves through 50 feet, W will be moved through 1 foot; but in moving P through 50 feet, 50 distinct efforts are made, by each of which 1 ounce is moved through 1 foot, and by which collectively 50 distinct ounces might be successively raised through 1 foot. But the weight W is 50 ounces, and has been raised through 1 foot; from whence it appears, that the expenditure of power is equal to that which would be necessary to raise the weight without the intervention of any machine.

This important principle may be presented under another aspect, which will perhaps render it more apparent. Suppose the weight W were actually divided into 50 equal parts, or suppose it were a vessel of liquid weighing 50 ounces, and containing 50 equal measures; if these 50 measures were successively lifted through a height of 1 foot; the efforts necessary to accomplish this would be the same as those used to move the power P through 50 feet, and it is obvious, that the total expenditure of force would be the same as that which would be necessary to lift the entire contents of the vessel through 1 foot.

When the nature and properties of the mechanic powers and other machines have been explained, the force of these observations will be more distinctly perceived. The effects of props and fixed points in sustaining a part of the weight, and sometimes the whole, both of the weight and power, will then be manifest, and every machine will furnish a verification of the remarkable proportion between the velocities of the weight and power, which has enabled us to explain what might otherwise be paradoxical and difficult of comprehension.

(227.) The most simple species of machines are those which are commonly denominated the MECHANIC POWERS. These have been differently enumerated by different writers. If, however, the object be to arrange in distinct classes, and in the smallest possible number of them, those machines which are alike in principle, the mechanic powers may be reduced to three.

1. The lever.
2. The cord.
3. The inclined plane.

To one or other of these classes all simple machines whatever may be reduced, and all complex machines may be resolved into simple elements which come under them.

(228.) The first class includes every machine which is composed of a solid body revolving on a fixed axis, although the name lever has been commonly confined to cases where the machine affects certain particular forms. This is by far the most useful class of machines, and will require in subsequent chapters very detailed development. The general principle, upon which equilibrium is established between the power and weight in machines of this class has been already explained in (183.) The power and weight are always supposed to be applied in directions at right angles to the axis. If lines be drawn from the axis perpendicular to the directions of power and weight, equilibrium will subsist, provided the power multiplied by the perpendicular distance of its direction from the axis, be equal to the weight multiplied by the perpendicular distance of its direction from the axis. This is a principle to which we shall have occasion to refer in explaining the various machines of this class.

(229.) If the moment of the power (184.) be greater than that of the weight, the effect of the power will prevail over that of the weight, and elevate it; but if, on the other hand, the moment of the power be less than that of the weight, the power will be insufficient to support the weight, and will allow it to fall.

(230.) The second class of simple machines includes all those cases in which force is transmitted by means of flexible threads, ropes, or chains. The principle, by which the effects of these machines are estimated, is, that the tension throughout the whole length of the same cord, provided it be perfectly flexible, and free from the effects of friction, must be the same. Thus, if a force acting at one end be balanced by a force acting at the other end, however the cord may be bent, or whatever course it may be compelled to take, by any causes which may affect it between its ends, these forces must be equal, provided the cord be free to move over any obstacles which may deflect it.

Within this class of machines are included all the various forms of pulleys.

(231.) The third class of simple machines includes all those cases in which the weight or resistance is supported or moved on a hard surface inclined to the vertical direction.

The effects of such machines are estimated by resolving the whole weight of the body into two elements by the parallelogram of forces. One of these elements is perpendicular to the surface, and supported by its resistance; the other is parallel to the surface, and supported by the power. The proportion, therefore, of the power to the weight will always depend on the obliquity of the surface to the direction of the weight. This will be easily understood by referring to what has been already explained in Chapter VIII.

Under this class of machines come the inclined plane, commonly so called, the wedge, the screw, and various others.

(232.) In order to simplify the development of the elementary theory of machines, it is expedient to omit the consideration of many circumstances, of which, however, a strict account must be taken before any practically useful application of that theory can be attempted. A machine, as we must for the present contemplate it, is a thing which can have no real or practical existence. Its various parts are considered to be free from friction: all surfaces which move in contact are supposed to be infinitely smooth and polished. The solid parts are conceived to be absolutely inflexible. The weight and inertia of the machine itself are wholly neglected, and we reason upon it as if it were divested of these qualities. Cords and ropes are supposed to have no stiffness, to be infinitely flexible. The machine, when it moves, is supposed to suffer no resistance from the atmosphere, and to be in all respects circumstanced as if it were in vacuo.

It is scarcely necessary to state, that, all these suppositions being false, none of the consequences deduced from them can be true. Nevertheless, as it is the business of art to bring machines as near to this state of ideal perfection as possible, the conclusions which are thus obtained, though false in a strict sense, yet deviate from the truth in but a small degree. Like the first outline of a picture, they resemble in their general features that truth to which, after many subsequent corrections, they must finally approximate.

After a first approximation has been made on the several false suppositions which have been mentioned, various effects, which have been previously neglected, are successively taken into account. Roughness, rigidity, imperfect flexibility, the resistance of air and other fluids, the effects of the weight and inertia of the machine, are severally examined, and their laws and properties detected. The modifications and corrections, thus suggested as necessary to be introduced into our former conclusions, are applied, and a second approximation, but still only an approximation, to truth is made. For, in investigating the laws which regulate the several effects just mentioned, we are compelled to proceed upon a new group of false suppositions. To determine the laws which regulate the friction of surfaces, it is necessary to assume that every part of the surfaces of contact are uniformly rough; that the solid parts which are imperfectly rigid, and the cords which are imperfectly flexible, are constituted throughout their entire dimensions of a uniform material; so that the imperfection does not prevail more in one part than another. Thus, all irregularity is left out of account, and a general average of the effects taken. It is obvious, therefore, that by these means we have still failed in obtaining a result exactly conformable to the real state of things; but it is equally obvious, that we have obtained one much more conformable to that state than had been previously accomplished, and sufficiently near it for most practical purposes.

This apparent imperfection in our instruments and powers of investigation is not peculiar to mechanics: it pervades all departments of natural science. In astronomy, the motions of the celestial bodies, and their various changes and appearances as developed by theory, assisted by observation and experience, are only approximations to the real motions and appearances which take place in nature. It is true that these approximations are susceptible of almost unlimited accuracy; but still they are, and ever will continue to be, only approximations. Optics and all other branches of natural science are liable to the same observations.