Chap. VIII.

GEOMETRY.

7 55

pyramids, having for their common altitude that of the frus-tum, and of which the bases are respectively the lower baseof the frustum, the upper base of the frustum, and a meanproportional between them.

A truncated Body is that of a pyramid, where L is socalled, when the top M is broken off.

Obelisks are of this form, and may have any number offaces, one of which is a triangle or other rectilineal figure,and the rest triangles which have a common vertex, and fortheir bases the sides of the first triangle or rectilineal figure :the altitude of such pyramids is the perpendicular distanceof the vertex from the base, and the truncated pyramid issaid to be triangular, quadrilateral, or pentagonal, accordingto the figure of its base. When the pyramid has a part ofits summit cut off by a plane parallel to its base, the partnext the base is called the frustum, and sometimes a trun-cated pyramid.

A Prism is a solid composed of many planes, of whichthe two opposite are equal; K, T, and S are instances oftriangular, square, and hexagonal prisms.

The prism is a solid contained by planes, of which twothat are opposite are equal, similar and parallel, and all therest are parallelograms. A right prism has its sides perpen-dicular to its ends ; an oblique prism is that of which thesides are oblique to the ends.

Tetraedron has four equal sides, formed by four equilateraltriangles, H and I, or it is a triangular pyramid having fourequal and equilateral faces, which are the least number pos-sible for a solid. If we assume a = the linear edge, 6 =the whole superficies, c = to the solid content, r = theradius of the inscribed sphere, r = the radius of the circum-scribed sphere, then the following relations hold true,a = 2r^ 6, 6 = 24 r* \/ 3, c = 8 r 3 V 3, R=3r.

Hexaedron is a solid which is bounded by six equal sides,as E and A. This is one of the five regular or Platonicsolids, the whole surface of which is equal to twenty-fourtimes the square of the radius of the inscribed sphere, and toeight times the square of the radius of the circumscribedsphere, and its solid content is eight times the cube of theinscribed sphere.

Dodecaedron is composed of twelve equal, equilateral, andequiangular pentagons, as the figure F.

The surface of the dodecaedron is found by multiplyingthe square of its side or linear edge into the number20-64578, and its solidity by multiplying the cube of itsside by 7-66312.

Icosaedron is a solid composed by twenty equal, equilateral,and equiangular triangles, as the figure G, and may beregarded as formed of twenty equal and similar triangularpyramids, whose vertices all meet at the same point; hencethe content of one of these pyramids multiplied by 20, givesthe whole content of the icosaedron.

ParaUelopipedon is a solid composed of six plane quad-rangles, of which the opposite sides are parallel, four ofwhich are equal to one another.

All the faces of one of the above named solids cannot beseen at the same time; in the cube A, we can from thepoint X only see the three faces A, D, and C, and the eyemay be so placed above it that only the top face may be seen.

In the representation of the several solids, it is necessarythat w*e should attend to some rules, and that they shouldalways be drawn, either as they appear to the eye, or ac-cording to a recognised scale. Isometrical perspective givesus, as we shall hereafter see, such a method of showing thevarious sides of a cube or other figure, and ascertaining froma scale its relative dimensions. When it is required that asolid should be projected in the plane of a picture with its

3c 2

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P H' y V O

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Fig. 748.

Fig. 749.

Fig. 750.

Fig. 751.

Fig. 753.