confequently, if feveral bodies are united in a ma chine, or feveral combinations of bodies to be fuf tained, no attention is to be paid to the center of gravity of the feveral parts which make the fyftem, but only to the center of the whole.

If a rod or beam be fo conftructed that the line drawn through the center of gravity and fuf. pention fhall be perpendicular to any given line in the rod, (for example, that which paffes along it's upper furface,) the rod hanging freely on it's axis of fufpenfion, will not be at reft except when that given line is in an horizontal position.

Sufpend this triangular piece, fig. 22, pl. 1, by the center a, and when it has ceafed vibrating, you will find the fide which is perpendicular to a line drawn through the center of gravity and fufpenfion is in an horizontal pofition.

That this must happen is evident, because the horizontal pofition, under the circumftances here defcribed, is the only pofition in which the line of direction paffes through the line of fufpenfion.

Upon this principle the ballance by which the weights of bodies are compared is conftructed,

OF THE SITUATION OF THE CENTER OF GRAVITY.

The center of gravity of any line is it's middle point.

The center of gravity in regular, uniform, and homogeneal folids, is at the center of it's figure.

For if from all the points on one fide of the furface of this folid, lines are drawn through the center to the oppofite fide, thefe lines will be divided into two equal parts by the center; the two parts of each line are equal, and of equal weight, and confequently the whole folid will be in equilibrio about that figure.

The

The center of gravity of the furface of a fphere, or regular polyhedron, is the center of the figure; that of a prifm or cylinder is in the middle of the axis that paffes through the center of gravity of their oppofite bafes.

To find the center of gravity of the surface of a triangle ABC, fig. 19, pl. 1, bifect any two of it's fides in D and E, draw lines from the oppofite angles to the points, and the center of gravity will be in the interfection of the lines; for EA bifects all the lines drawn parallel to BC, which conftitute the furface of the triangle, therefore it paffes through the center of gravity of each line, and for the fame reason DC paffes through thern also.

A fimilar reafon proves, that the center of gravity of a regular polygon, whofe number of fides is an odd number, is the fame with the center of the figure.

The center of gravity of a triangle may be found by taking two-thirds of the line drawn from any one of it's angles to the middle of it's oppofite fide.

To FIND THE CENTER OF GRAVITY OF A

TRAPEZIUM.

Find the center of gravity of the two triangles formed by a diagonal, and join them by a trait line; find and join alfo by a ftrait line the center of gravity of two triangles formed by another diagonal; the interfection of thefe two lines will be the center required.

By the fame means we may cafily find the center of gravity of any polygon.

Divide the polygon into as many triangles lefs two as it has fides, find the center of gravity of each triangle, and then confidering thefe centers

as

as fo many bodies, find the common center of the whole fyftem.

TO FIND THE CENTER OF GRAVITY OF A PYRAMID.

It is evident, ift, That the elements of a pyramid are polygons fimilar to the base, and fimilarly fituated, whofe dimenfions decrease in an arithmetical progreffion from the base to the summit; therefore the ftrait line which goes from the center of gravity of the bafe to the fummit, will pass through the center of gravity of all these elements, and confequently by that of the pyramid. 2d, If the pyramid is triangular, each face may be taken for the bafe, and the oppofite angle for the fummit; whence it follows, that the center of gravity of a triangular pyramid S A B C, fig. 21, pl. 1, is at the interfection of the two right lines S F, A E, drawn from the two angles S, A, to the centers of gravity of the oppofite faces, or of the length of each line S D, AD, drawn on the plane of the oppofite face from the angles S, A, to D, the middle point of the fide of the base oppofite to these angles.

The center of gravity of a cone or pyramid is of the axis, reckoning from the fummit or

vertex.

For the center of gravity of the fector of a circle, fay as the arch to it's chord, fo is of radius to the distance of the center of gravity from that of the circle.

For an arch of a circle, as the arch to the fine of the arch, fo is radius to the distance of 1/4 the center of gravity from the center.

Thus in bodies that are of uniform denfity, and at the fame time admit of geometrical menfuration, the pofition of the center of gravity may be afcertained

afcertained by theory; but if they be wholly irregular, recourfe must be had to experiment.

As any body confidered in mechanics is only an aggregate of feveral other bodies or parts, fo the center of gravity of a body is only the common center of gravity of all it's parts; and confequently, if feveral bodies are united in any machine, or if there be any combination of bodies to be sustained, we are no longer to regard the particular centers of gravity of the compound, but only the common center of gravity of the whole.

Thus a windmill fhould be fupported under the common center of gravity of all it's parts, and it's line of direction fhould coincide with the axis of the poft round which it moves; and a crane on a wharf or a dock, where the whole machine turns round, fhould have the line of direction in it's axis.

Let the line A B, fig. 23, pl. 1, represent an even rod or wire divided into two equal parts at the point e, it's center of gravity will be at C. If two equal bodies be fixed upon the ends thereof, fo as to have their centers of gravity at the fame distance from C, they will be in equilibrio about the faid point, which will become their common center of gravity, and continue fo, whether the bodies approach to or recede from it, in proportion to their maffes.

The fame will happen if the bodies are unequal, as A and b, fig. 24, pl. 1, whofe maffes are to each other as two to one, provided the greater body at A be twice as near the common center of gravity as the leffer b; and c will be the common center of gravity of thofe bodies, though they fhould move to immense distances from each other, provided their distances from the faid point are reciprocally as their maffes.

So that when two bodies approach to or re

cede

cede from each other with velocities reciprocally proportional to their maffes, their center of gravity will remain at reft.

If the bodies be made faft upon the wire, and the center of gravity be fuftained on a pivot, it will remain at reft though the bodies revolve round with the utmoft velocity; and the bodies will defcribe fimilar circles about it and about each other, the one never overpowering the other.

If they be carried forward in any manner by any external force acting upon them in proportion to their maffes, their center of gravity will go forward uniformly in a strait line, and be moved just as if the two bodies were united into one at that center; and if they be projected, their center of gravity will move in the fame curve as other projectiles, which is evident by the motion of an arrow, of chain fhot, and of a flick thrown from the hand, the center of gravity of thefe bodies moving like a fingle ball.

So alfo the earth and moon in their motion round the fun, do neither of them defcribe the real regular orbit, but it is defcribed by their common center of gravity in the fame manner as if they were both united in that point, or in the fame manner the earth is fuppofed to do when thefe inequalities of motion are overlooked; and if their dif tances from the common center of gravity be reciprocally proportionable to their males, their diftances from each other may be greater or lefs in any proportion.

If to the two bodies A and B, fig. 9, pl. 2, there be added a third at D equal to one of the other, let A and B be reduced to their common center of gravity, and be confidered as a body equal to both placed at C, then the common center of gravity of C and D will be found at K, as much nearer to C, as the mafs of the body or bo

dies