()6 LECTURE VII. 



a force equivalent to two ounces, drawing the lever upwards ; and instead of 

 one of the weights, we may place the end of the lever under a firm obstacle, 

 and the equilibrium will still remain, the lever being now of the second kind. 

 Here therefore, the weight remaining at the other end of the lever counter- 

 balances a force of two ounces, acting at half the distance from the new ful- 

 crum ; and we may substitute for this force a weight of two ounces, acting at 

 an equal distance on the other side of that fulcrum, supposing the lever to be 

 suihciently lengthened, and there will still be an equilibrium. In this case the 

 fulcrum will sustain a weight of three ounces; and we may substitute for it a 

 force of three ounces acting upwards, and proceed as before. In a similar 

 manner the demonstration may be extended to any commensurable proportion 

 of the arms, that is, any proportion that can be expressed by numbers ; and 

 it -is easy to show that the same law must be true of all ratios whatever, even 

 if they happen to be incommensurable, such as the side of a square, compared 

 to it§ diagonal, which cannot be accurately expressed by any numbers what- 

 ever; the forces remaining always in equilibrium, when they are to each other 

 inversely as the distances at which they are applied. 



It is sometimes more convenient to have a series of levers acting on each 

 otlier, with a moderate increase of power in each, than to have a single lever 

 equivalent in its effect. We may also bend either arm of a lever in any man- 

 ner that we please, without altering its power, provided that the direction of 

 the force be perpendicular to the line drawn to the fulcrum; or if the force be 

 applied obliquely, it may always be imagined to act at the end of a lever equal 

 in length to the perpendicular let fall from the fulcrum on the direction of the 

 force. Thus, if two levers are connected by a rope or bar, when the direction 

 of one of them nearly coincides with that of the rope, a forc'G applied trans- 

 versely to the lever acts with a great mechanical advantage against the rope ; 

 but as the inclination increases, the advantage gradually diminishes, and 

 changes, at last, to an ecpial advantage on the side of the rope and the other 

 lever to which it is attached. When therefore a great force is required in the 

 beginning of the motion, and afterwards a much smaller force with a greater 

 velocity, this apparatus may be extremely convenient: thus, in opening a 

 steam valve, the pressure of the steam is at first to be overcome, and after this, 

 little or no additional force is required ; and j\Ir. Watt has very ingeniously 

 applied this arrangement of levers to the purpose in his steam engines. In the 



