776

THEORY AND PRACTICE OF ENGINEERING.

The Construction of Solids comes more under the denomi-nation of descriptive geometry, and on the Continent it hasfor many years formed a branch of study for engineers, bothcivil and military : it cannot be too highly esteemed, as itconsists in the application of all the known rules of projec-tion, to exhibit on a plane the figures of the solids, as wellas to show their method of construction. The plane surfacesof all solids are bounded by edges, which can be expressedby straight lines: and in the construction of a solid we haveto regard three varieties of angle: the first are those wherethe lines meet which bound the figure; the second, thosewhich result from several faces meeting to form a solidangle; and thirdly, those which are formed by two planesor faces.

We shall find that cubes contain six equal planes, twelveedges, and eight solid angles, and that in all solids withplane surfaces, the edges terminate in solid angles formedby them, or where they unite with each other : and to findthe projections of the right lines which represent those edges,it is necessary that we should know the position of the solidangles where they meet, and these arc formed generally ofseveral plane angles.

To make a Triangular Pyramid. , draw the triangle D E Fof the required dimensions, and then fix the point G wherethe summit is required, and unite lines from each of thepoints D E F of the base in this point, which will give thefigure required. All other pyramids, as those with squareor polygonal bases, are set out in the same manner.

Pyramids may be regarded as solids standing on polygonalbases, their planes or faces being triangular, and meeting ina point at the top, where they form a solid angle.

To construct a Pyramid or a Tetraedron in relief, in card, orother material, the base must be set out as at I; then at thesides the other triangles must be formed. These three outertriangles are then raised and united at the top, which willform the tetraedron required. If the faces which meet at theapex are required to be longer than the equilateral tri-angle, after the base is set out, the isosceles triangle must betraced of the height required, and when cut out, united atthe point or summit as before.

It will easily be seen that a tetraedron may be inscribedin a tetraedron, an octaedron in a cube, and a cube in anoctaedron, an icosaedron in a dodecaedron, and a dode-caedron in an icosaedron.

The mutual relation between the regular solids is verycurious: when lines are drawn from the centre of the cir-cumscribed solid to its different angular points, these lineswill be perpendicular respectively to the faces of the inscribedsolid; so tiiat if we cleave or cut away the solid angles of thecircumscribed figure by planes perpendicular to these lines,and if we continue the process until we arrive at the centresof the several faces, we shall obtain the regular solid, whichis inscribed, and which forms, as it were, the nucleus of theother. I3y cutting away the solid angles of the tetraedron,we also form the octaedron.

To find the inclination of the two adjoining planes of atetraedron, we have only to consider that the required in-clination is that of two angles of equilateral triangles, which,together with a third, form a solid angle, and therefore maybe easily constructed : in a cube the angle of inclinationwill be a right angle.

To draw or construct Prisms -Set out its base of the

number of sides given ; then from the angles of the baseDCH, &c., elevate perpendiculars of the height to be givento the prism, in such a manner that we join the pointsJ.C,N, &c. Hollow prisms may also be set out, by givingthe thickness and drawing as it were one prism within theother.

G

E

D

Fig. 851.

Fig. 852.

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0

Fig. 853.