. Spons' dictionary of engineering, civil, mechanical, military, and naval; with technical terms in French, German, Italian, and Spanish . en we were seeking the centre of gravity of the tetra-hedron. Any Pyramid. — Let SABCDE, Fig. 3334, be a pyramid withany base. Decompose it into tetrahedrons by the diagonal planesASC and A S D. At a distance from the base equal to a quarterof the height of the pyramid, draw a plane abed e parallel to thisbase. This plane will contain the centres of gravity g, g, g, of thepartial tetrahedrons, and consequently the centre of gravity of thewhole pyramid (Prin
. Spons' dictionary of engineering, civil, mechanical, military, and naval; with technical terms in French, German, Italian, and Spanish . en we were seeking the centre of gravity of the tetra-hedron. Any Pyramid. — Let SABCDE, Fig. 3334, be a pyramid withany base. Decompose it into tetrahedrons by the diagonal planesASC and A S D. At a distance from the base equal to a quarterof the height of the pyramid, draw a plane abed e parallel to thisbase. This plane will contain the centres of gravity g, g, g, of thepartial tetrahedrons, and consequently the centre of gravity of thewhole pyramid (Principle I). But the tetrahedrons S A B C, S A C D,SADE, having the same height, are to each other as their bases, oras the triangles abc, a cd, ade, proportional to these bases. There-fore, if we suppose applied to the points g, g, g, weights equal tothose of the corresponding tetrahedrons, these weights would be at the same time proportional tothe areas of the triangles abc, a cd, ade. Hence it follows that the point of application of theresultant of these weights is no other than the centre of gravity of the polygon abc de. But it. GKAYITY. 1713 may be easily seen by simple similitudes of triangles, that the straight line which joins the summit Sto the centre of gravity of the base A B C D E of the pp-amid, passes through the centres of gravityof all the sections, as abc de, parallel to this base. Therefore the centre of gravity of any pyramidis upon the straight line vMch joins the summit to the centre of gravity of the base^ at a distance of one-fourth of this line from the base. This theorem extends to a cone, whether right or oblique, and with any base, since such a bodyis a pyramid whose base is a polygon with an infinite number of infinitely small sides. Truncated Pyramid.—If the frustum of the pyramid be decomposed into frusta of tetrahedrons,their upper bases will be proportional to the lower, and generally to the sections made by the sameplane parallel to the bases
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