(241.) The sense in which a small power is said to sustain a great weight, and the manner of accomplishing this, being explained, we shall now consider how the power is applied in moving the weight. Let P W, _fig. 81._, be the places of the power and weight, and F that of the fulcrum, and let the power be depressed to P′ while the weight is raised to W′. The space P P′ evidently bears the same proportion to W W′, as the arm P F to W F. Thus if P F be ten times W F, P P′ will be ten times W W′. A power of one pound at P being moved from P to P′, will carry a weight of ten pounds from W to W′. But in this case it ought not to be said, that a lesser weight moves a greater, for it is not difficult to show, that the total expenditure of force in the motion of one pound from P to P′ is exactly the same as in the motion of ten pounds from W to W′. If the space P P′ be ten inches, the space W W′ will be one inch. A weight of one pound is therefore moved through ten successive inches, and in each inch the force expended is that which would be sufficient to move one pound through one inch. The total expenditure of force from P to P′ is ten times the force necessary to move one pound through one inch, or what is the same, it is that which would be necessary to move ten pounds through one inch. But this is exactly what is accomplished by the opposite end W of the lever; for the weight W is ten pounds, and the space W W′ is one inch. If the weight W of ten pounds could be conveniently divided into ten equal parts of one pound each, each part might be separately raised through one inch, without the intervention of the lever or any other machine. In this case, the same quantity of power would be expended, and expended in the same manner as in the case just mentioned. It is evident, therefore, that when a machine is applied to raise a weight or to overcome resistance, as much force must be really used as if the power were immediately applied to the weight or resistance. All that is accomplished by the machine is to enable the power to do that by a succession of distinct efforts which should be otherwise performed by a single effort. These observations will be found to be applicable to all machines whatever. (242.) Weighing machines of almost every kind, whether used for commercial or philosophical purposes, are varieties of the lever. The common balance, which, of all weighing machines, is the most perfect and best adapted for ordinary use, whether in commerce or experimental philosophy, is a lever with equal arms. In the steel-yard one weight serves as a counterpoise and measure of others of different amount, by receiving a leverage variable according to the varying amount of the weight against which it acts. A detailed account of such instruments will be found in Chapter XXI.