ON PRESSURE AND EQUILIBRIUM. . 51 



manner the demonstration may be extended to any commensurable propor- 

 tion 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 what- 

 ever, even if they happen to be incommensurable, such as the side of a 

 square compared to its diagonal, which cannot be accurately expressed by 

 any numbers whatever ; 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 

 other 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 manner 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 ful- 

 crum 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 force applied transversely 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 equal advantage on the 

 side of the rope and the other lever to which it is attached. When, there- 

 fore, a great force is required in the beginning of the motion, and after- 

 wards 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 Mr. Watt has very ingeniously applied this arrangement 

 of levers to the purpose in his steam engines. In the same manner, it is 

 necessary that the platten of a printing press, or the part which presses the 

 paper on the types, should descend from a considerable height, but it is 

 only at the instant of taking off the impression that a great force is re- 

 quired ; and both these ends are obtained by similar means in a press 

 lately invented by Lord Stanhope. (Plate III. Fig. 48, 49.) 



The wheel and axis bear a very strong resemblance to the lever. If two 

 threads, or perfectly flexible and inextensible lines, be wound in contrary 

 directions round two cylinders, drums, or rollers, moveable together on 

 the same axis, there will be an equilibrium when the weights attached to 

 the threads, or the forces operating on them, are inversely as the radii of 

 the cylinders, or as the diameters of which they are the halves. It may 

 easily be understood that the weights have the same power in turning 

 round the cylinders, as if they were immediately attached to the arms of a 

 lever equal in length to their semidiameter, and that the conditions of 

 equilibrium will be the same. The demonstration may also be more im- 

 mediately deduced from the position of the centre of gravity immediately 

 below the axis of the cylinders, which requires, the weights to be inversely 

 as the radii. With respect to stability, the equilibrium is neutral, and the 

 cylinders will remain at rest in any situation. A single cylinder is also 

 often combined with a lever or winch, and in this case the radius of the 

 cylinder is to be compared with the length of the lever or winch. (Plate 

 III. Fig. 50.) 



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