8 PHYSICS. 
13), obtain first the centres of gravity, S and s, of the two, by means of 
diagonals, and unite the two points by the straight line, Ss ; upon this latter 
determine the centre of gravity, S’, as will be explained more fully under the 
head of the lever. The same method is to be pursued in determining the 
centre of gravity of irregular surfaces, as for instance, ABCD (jig. 9). 
b. Of Simple Machines. 
Simple machines, or mechanical powers, are those simple arrangements of 
which all machinery is compounded. Of these, six are generally distin- 
guished: the lever, the wheel and axle, the pulley, the inclined plane, the 
wedge, and the screw. All these, however, may strictly be reduced to two— 
the lever and the inclined plane; on which account these two are looked 
upon as the elementary machines. The ancient Greek mathematician, 
Pappus, enumerates the above-mentioned simple machines, with the excep- 
tion of the inclined plane, which is of more recent introduction. Instead 
of the latter power, Varignon added the funicular machine to the five others, 
which, however, consisting simply of ropes on which the forces act in 
different directions, and being intended to elucidate the proposition of the 
composition of forces, cannot properly be called a simple machine. See 
jig. 14, where the forces act in the same plane and in different directions 
upon the combined ropes at A, E, P, P’, P’”. These will hold each other in 
equilibrium when BC is equal and opposite to the mean force of BA and BP’, 
CD equal and opposite to the mean force of DE and DP’”, and CP equal 
and opposite to the mean force CB and CD. 
The mathematical lever, in its simplest form, is an inflexible line sup- 
ported in one point ( fulcrum, hypomochlium) on which two or more forces 
operate, endeavoring to move it about this fulcrum. The distances from 
the fulcrum to the points of attachment of the forces are the arms of the 
lever. There are two kinds of levers: levers of the first class, or double- 
armed levers, in which the forces operate on different sides of the fulcrum ; 
and levers of the second class, or one-armed levers, in which these act on 
the same side. The same conditions of equilibrium, however, apply to both, 
viz. that the forces must be inversely as the arms of the levers. Thus, 
when the arms of the lever are equal, the forces must be equal, and when 
the arms are unequal, the forces must be unequal, the greater force acting 
on the shorter arm, and the lesser force on the longer arm, these forces 
being in the same proportion as the arms of the lever. PZ. 16, fig. 28, 
represents a lever of the first class, in which the acting forces are 
the weights, P and W. F is the fulcrum, and for equilibrium, the pro- 
portions P: W::BF: AT’ must exist. Mig. 25 represents a lever of the 
second class, which is supported at EF’, and operated upon in opposite 
directions by the weight W and the weight P, passing over the pulley and 
attached to A, the former weight drawing the lever downwards, the latter 
raising it up. Equilibrium can only subsist when P: W:: BF: AF. Fig. 
26 is properly a lever of the second class, although in it the fulcrum is 
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