MECHANICS. 7 
CE that of W’. Pl. 16, fig. 16, extends this construction to the case of 
several weights, and forms the basis of the Funicular Machine of Varignon, 
of which more hereaftei 
It is known that every body is subject to the influence of gravitation, and 
that this gravitation acts upon every molecule of the body. All these single 
influences of gravitation may be considered as united into a mean force of 
gravitation, which then is called the weight of the body. This union can 
and must take place in a single point, the centre of gravity; and a force 
acting on this centre of gravity, and equal to the weight of the body, will 
hold it in equilibrium. Gravity and weight, therefore, difler as cause and 
effect. Gravity is that natural force which causes the weight of bodies, and 
the centre of gravity the point in which the entire weight of the body may 
be supposed to reside. It is a fixed point, whose situation does not change, 
whatever be the position of the body. Whenever this point is supported in 
any way, the body rests in equilibrium. 
The centre of gravity of homogeneous bodies of regular shape, is easily 
obtained by geometrical constructions. The centre of gravity of a straight 
line is evidently at its middle point (fig. 5). That of a triangle, ABC (fig. 
6), lies where lines drawn from the angles to the centres of the opposite sides 
intersect each other. It may also be found by drawing a line from one 
angle to the middle of its opposite side, and trisecting this line; the first 
point of division, S, starting from D, will then be the centre of gravity. 
That DS must equal 4DB, is shown by drawing DE; DE will evidently 
=3AB. The triangles DSE and ASB are, however, similar, whence 
SD:SB::DE: AB; as, however, DE = }AB, SD must = 3SB = 4DB. 
The centre of gravity, 8, of a parallelogram, ABCD ( fig. 8), is the inter- 
section of its diagonals; that of a tegular polygon, ABCDEF ( fig. 7), as also 
of a circle, is the centre. Ifa rectilineal figure of an even number of sides, 
as, for instance, the six-sided one, ABCDEF ( fig. 7), be so constituted as to 
be divisible by a diagonal, CF, into two symmetrical halves. the centre of 
gravity will lie in the middle of this diagonal. If, moreover, as in the figure, 
all diagonals have a common point of intersection, this point itself will be 
the centre of gravity. 
In those bodies which have a regular shape, and whose mass is distributed 
with perfect uniformity, the centre of gravity may be likewise determined 
geometrically. Thus, the centre of gravity of a cube or parallelopipedon is 
also in its geometrical centre: it is obtained either by passing a plane 
through two opposite edges, AB, DE ( pi. 16, fig. 10), and finding the centre 
of this plane, or by finding the centres of gravity, S, S’ (fig. 11), of two 
opposite planes, and bisecting the connecting line at S’. From the first 
method it follows that the centre of gravity of a parallelopipedon lies in the 
point of intersection of two of its diagonals. 
The centre of gravity of a pyramid (fig. 12) is obtained by connecting 
the apex, G, with the centre of gravity of the base, S, and on this line 
cutting off the fourth part from the base, so that SS’= GS. The centre of 
gravity of the cone is found in a similar manner. To obtain the common 
centre of gravity of two different bodies, as of the cubes AG and ag (fig. 
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