205. We may combine the cases of Art. 201 and Art. 203 into one statement thus: P must be to W in the same proportion as the distance of W from the fulcrum is to the distance of P from the fulcrum. Here the distance of W from the fulcrum means the length of the perpendicular from the fulcrum on the line of action of W; and the distance of P from the fulcrum is to be understood in a similar way. The same statement will hold with respect to the equilibrium of a Lever, whether it be straight or bent.

206. What we have thus obtained with respect to Levers of the first class holds also for Levers of the second and third classes, straight or bent. Thus, universally, in order that there may be equilibrium on a Lever, P and W must tend to turn the Lever in contrary directions, and P must be to W in the same proportion as the distance of W from the fulcrum is to the distance of P from the fulcrum.

207. There is no difference in theory between Levers of the second class and Levers of the third class, but there is considerable difference in practice. For it follows from the statement of Art. 206 that in Levers of the second class the Power is less than the Weight, and that in Levers of the third class the Power is greater than the Weight. Thus we may say that there is a mechanical gain by using a Lever of the second class, and a mechanical loss by using a Lever of the third class. The advantage of a machine may be defined as the proportion of the Weight to the Power when there is equilibrium.

208. It follows from the statement of Art. 206 that a very small Power might be made to balance a very great Weight by using a suitable Lever, that is by making the distance of the Power from the fulcrum very large, and the distance of the Weight from the fulcrum very small. But machines are in general used rather to produce motion than to prevent it; and this leads to a very important remark. Suppose in the diagram of Art. 201 that BC is one-third of AC, then when there is equilibrium the Power is one-third of the Weight. It is true then that a force just exceeding one-third of the Weight will be sufficient to move the Weight; but on the other hand it will be found

that if the Weight is to be raised through one inch the Power end of the Lever must descend through three inches. Thus although by the aid of a Lever we can move any Weight by a force much less than that Weight, yet the force must be exerted through a distance which is greater in a corresponding degree. This important principle is found to apply to the other Mechanical Powers, and to combinations of them, and it is usually stated briefly thus: what is gained in power is lost in speed.

209. The Principle of the Lever which we have explained was first demonstrated by Archimedes, the greatest mathematical philosopher among the ancients, and probably inferior to Newton alone among the moderns. Tradition has recorded in a well-known sentence attributed to him the high opinion which he had formed of the importance of his result: shew me where I may stand and I will move the world. A more tolerable form of the boast would be, I will support the world; for in order to move the world through an appreciable distance, the philosopher by the principle of Art. 208 would have had himself to move through an enormously greater distance. We now know that if a motion of the earth through an infinitesimal space is all that is required we may dispense with the Lever which Archimedes proposed to use, for the motion is produced every time a man jumps from the ground: he really pushes the earth from beneath him by his spring, and then draws it towards him by his weight. See Chapter VIII.

210. It is unadvisable to introduce more technical terms than are absolutely necessary into an elementary work of the present kind; but one such term may be noticed by the aid of which the Principle of the Lever can be briefly stated. The Moment of a force with respect to a point is the product of the force into the perpendicular from the point on the line of action of the force. This supposes the force to be expressed in pounds, or ounces, or in any other convenient terms; and the perpendicular to be expressed in inches, or feet, or any other convenient terms. But having once chosen the unit of force, and the unit of length, we must keep to these units throughout the investigation on which we may be engaged. With the aid

of the term moment we may express the relation which must hold between the Power and the Weight for equilibrium on the Lever thus: the moments of the Power and the Weight with respect to the fulcrum must be equal. It is easy to see that this coincides with what is stated in Art. 206.

211. In the present Chapter we have left out of consideration the fact that the rod or bar of the Lever will itself have weight. If the fulcrum be at the centre of gravity of this rod or bar the weight of the rod or bar is entirely supported by the fulcrum, and so need not be regarded; the rod or bar so far as we are concerned with it is practically without weight. But if the fulcrum is not at the centre of gravity of the rod or bar, allowance must be made for the weight of the rod or bar: the account of the Common Steel-yard in the next Chapter will illustrate this point.

XIII. THE BALANCE.

212. The various kinds of Balances form such a very important application of the Principle of the Lever that we shall devote a separate Chapter to them. The use of the Balance, as is well known, is to determine the weight of any proposed body, so that in this case we employ the Lever not to produce motion, but to prevent motion, that is to preserve equilibrium.

213. The Common Balance.

The Common Balance consists of a beam with a scale suspended from each end; the beam can turn about a fulcrum which is above the centre of gravity of the beam, so that if the scales were removed the beam would adjust itself to a position of stable equilibrium: see Art. 183. The arms of the beam should be of equal length, and the scales of equal weight, so that the beam may be at rest in a horizontal position when the scales are attached and are empty. If these conditions are satisfied the Balance is said to be true; if not it is said to be false. The body to be

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weighed is placed in one scale and weights in the other until the beam remains at rest in a horizontal position. In this case if the Balance be true the weight of the body is indicated by the weights which have been put in the other scale. We may test whether the Balance is true by observing whether the beam still remains at rest in a horizontal position when the contents of the two scales are interchanged. But even if a Balance be false we may determine by its aid the exact weight of a body, if we employ the process which is called double weighing. Put the body which is to be weighed in one scale, and in the other scale put sand or shot so as exactly to counterpoise the body. Remove the body and put in its place weights so as just to restore equilibrium again. Then the sum of these weights indicates the weight of the body. This process of double weighing is very simple in theory and very exact in practice.

214. Another kind of Balance is that in which the arms are unequal, and the same Weight is used to weigh different substances by putting it at different distances from the fulcrum. The Common Steel-yard is of this kind.

Let AB be the beam of the Steel-yard, C the fulcrum. Let A be the fixed point from which the body to be weighed is suspended. Let Q be the weight of the beam together with the hook or scale-pan at A. Let P be a

weight which may be placed at any distance from the fulcrum. We have now to graduate the Steel-yard, that is to put marks on it so that if we observe the position which P has when a body is suspended from A, and the whole is in equilibrium, we may know the weight of that body. Now we might proceed by the aid of theory. For the weights P and Q being parallel forces we can determine their resultant by Art. 165; and then this resultant must balance the weight of the body, according to the Principle of the Lever. But it will be more simple to proceed by the aid of experiment. Take then a weight, say of one pound, and suspend it from A; move P about until such a place is found for it that the beam just remains in equilibrium, and mark the place with the figure 1. Again instead of the weight of one pound at A put a weight of two pounds; move P about as before, and mark with the figure 2 the place which it has when the beam is in equilibrium. Proceeding in this way the beam becomes graduated, and the Steel-yard is fit for use. It will be found by trial that the figures 1, 2, 3, 4, ... succeed at equal distances on the beam. Thus when we have a body to be weighed we suspend it from A, and then move P about until it comes to such a place that the beam remains at rest in a horizontal position. Let this, for example, be when P is midway between the figures 3 and 4, as in the diagram; then we infer that the body weighs 3 pounds.

216. Sometimes two different graduations are recorded on the Steel-yard, corresponding to two different moveable Weights. In this way we can extend the range of the machine without making the machine itself inconveniently long; thus one graduation might give us the weights of bodies up to 10 pounds, and then another graduation corresponding to a heavier moveable weight might give us the weights of bodies of 10 pounds and upwards to about 100 pounds. Or the two graduations may correspond to two different positions of the point A from which the body to be weighed is hung, the moveable weight P being the same in both cases.

217. Another kind of Steel-yard is called the Danish Steel-yard.