MECHANICS. I] 
MARYS , 
and to produce equilibrium, ee 2 EY Must == — pt OF Fo atk, 
ALE. AN = Wi EY . BURY, 
The lever considered thus far has been the mathematical or weightless 
one; in practice, however, its weight must be taken into account as acting 
at its centre of gravity. Calling, therefore, the weight of the lever Q, and 
the distance of the centre of gravity from the fulcrum, q, the conditions of 
equilibrium in fig. 23 will be P.FA+Qq=W.FB; for figs. 25, 26, and 
30, P.FA=W.FB+Qq; for fig. 29, P.cos.c.FA=W cos. 6. FB+ 
Qq; and for fig. 27, P.FA+P’. FA’ + P’. FA"V=W.FB+ W’. FB’ + 
WEB’ + Qg. 
The general principles of the rectilineal lever apply to the case of bent 
levers, or those whose arms form an angle with each other at the fulcrum. 
Here, however, equilibrium is established when a line drawn from the ful- 
crum, perpendicular to the straight line connecting the extremities of the 
lever, divides this line into two parts which are inversely proportional to 
the forces acting on the ends of the lever. The bent lever is much more 
sensitive than the straight, when its angle is directed upwards, for which 
reason, in the better scale-balances, the beams are not rectilineal levers, but 
the fulcrum or point of suspension is generally somewhat lower than the 
points of attachment of the weights. 
To the preceding proportions respecting the lever, it becomes necessary 
to add, that in every lever, the spaces traversed by the arms of the lever are 
inversely as the weights or forces, and directly as the lengths of the arms, 
so that when, for instance, the arms are as 1:3, the spaces traversed will be 
1:3. This proposition is of great importance, as it follows from it that by 
an elongation of the arm of the lever to which the power is applied, the 
effect of the lever may be increased in proportion, but that the time required 
for the production of a particular effect is also increased; so that what is 
gained in power is lost in time. Archimedes, after developing the law of 
the lever, was correct in saying, “ Give me a fulcrum out of the earth and I 
will raise her from her foundations.” But let us see what effort it would 
cost him. Supposing him to work for ten hours each day, and to exert a 
force of 30 pounds in pulling an arm of the lever through 10,000 feet per 
hour, he would, in the space of 1,473,973,790 centuries, have elevated the 
earth just one inch! Tor, let the force exerted = 30 lbs., the weight of 
the earth =W, and the arc described by the long arm of the lever in moving 
the short arm one inch = 2, then 80 X x=W x1, and z ee that is, to the 
entire weight of the earth divided by 30. 
Now, supposing the earth to be a sphere of a mean radius = 3949 miles, 
then, since the volume of a sphere — = 4R%), the earth will contain about 
256,827,726,120 cubic miles. Asa cubic mile of water, at the rate of 621 lbs. 
to the cubic foot, will weigh 1,752,400,000 lbs., and as the mean density of 
the earth, according to Cavendish, is 53 times that of water, the cubic mile 
of earth will weigh 5! times this amount, or 7,688,200,000 lbs. The entire 
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