THE LEVER AND WHEELWORK. 



245 



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 ob- 

 vious 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 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 pro- 

 portion between the velocities of the weight and power, which has enabled us 

 to explain what might otherwise be paradoxical and difficult of compre- 

 hension. 



The most simple species of machines are those which are commonly denom- 

 inated the mechanic powers. These have been differently enumerated by differ- 

 ent 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 prin- 

 ciple, 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 re- 

 duced, and all complex machines may be resolved into simple elements which 

 come under them. 



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. 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 wefght, equilibrium will subsist, provided the power, multiplied by the per- 

 pendicular distance of its direction from the axis, be equal to the weight multi- 

 plied 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. 



If the moment of the power 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. 



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. 



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 \ 



