The octahedron or double four sided. pyramid, figs. 589. 590, has four of its centers upon the lines e f, and four upon g h; the positions for these circles upon either side of the circumference, being determined by the line e h, at 35 degrees* to C D, first drawn on paper as before. One of these circles g h, having been marked on the ball with pencil, it, and the circumference C. D, are divided into four by the division plate; the sphere is then reversed in the plain chuck and the other circle e f, marked and divided into four, the index being adjusted that the four marks may exactly agree with those on C. D, that thus every pair of centers may be exactly opposite each other upon the same four lines. The filling pieces, fig. 590, are similar to those already described.

The division of the surface of the sphere for the centers of the twelve equilateral pentagons forming the dodecahedron, perhaps the most beautiful solid of the group, is also that which is most useful for other spherical works and ornamentation. The form and arrangement of its faces are indicated in fig. 591. The circle, the diameter of the sphere, and the line e h, drawn at 26 degrees† to the circumference C. D, give the distance for marking the work with two circles e f, and g h; each of these two circles is then divided into five equal parts by the division plate, but alternately; so that the points marked on the one circle, are opposite the spaces and exactly between the marks on the other, this places ten centers, the remaining two being at A. and B. in the axial line, and as in the previous examples the end way of the grain.

The centers of the twenty equilateral triangles of the icosa-hedron, fig. 592, lie in pairs, alternately upon the circles a b, c d, and e f, g h; found by the lines a h, and c f, respectively at 52 degrees‡ and 10 degrees § to C. D, by the division of all those circles into five equal parts. The circles a b, and c d, being divided alike, and the circles e /, and g h, being also divided alike; but the points of division on the two former circles being intermediate to those in the two latter. The manipulation in turning these solids is in every respect that already described, but the increased number, requires greater care in marking and adjusting the centers to run true, to ensure that all the faces are true planes, that they are all reduced to a precisely similar depth from the surface of the sphere, and that all the solid angles are left perfect.

* Mathematically 35° 15' 52". ‡ Mathematically 52° 37' 21".

† Mathematically 26° 33' 54". § Mathematically 10° 48' 44".

The separation of the solid polyhedra into several shells, contained one within the other, is accomplished much after the same manner as with the Chinese ball, but with straight in place of curved tools and guide. Every external face at its completion upon the sphere, and before proceeding to form the next, is pierced with a radial aperture flat at the bottom, proportioned in depth to the number of shells and their intervals, or the widths of the separating tools, and in diameter, to the length of the blades and the breadth of material to be severed. The exact form requisite may be previously set out on paper and a template made to it, similar to fig. 567, that all the apertures in the different faces may be turned to one size. As every aperture is made, it is fitted with a stopper, fig. 593, bearing upon the bottom of the hole, which will be one face of the central solid; the head, turned to the curve of the sphere to serve as the filling piece, and cut with an internal screw to insert a handle to assist its removal.

Fig. 591. Fig. 592. Fig. 593.

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The separating tools are used in a guide similar to fig. 566, but with the front, which bears upon the flat faces of the solid, straight and parallel with the blade of the tool. The face of the central solid is first separated and then the shells, from the most central outwards, after which the stopper is inserted in its place; this is retained in position by fitting the aperture, but requires securing, by slightly cementing under its head. The other faces are then proceeded with in the same manner, but, as every one becomes inaccessible, so soon as its stopper is inserted, the similarity of depth to which the different separating tools are placed from the surface, requires great care. Accuracy in this respect is however readily attained, by employing a distinct tool and guide for every shell; these tools remaining fixed in their guides, as they were set for turning the different faces at the first aperture. It should be observed that the exterior faces and angles of each shell, are copies of the external solid, but that their internal surfaces are not entirely so, for the superficies of the separating grooves being circular, they cannot quite approach the internal angloids; as these however are not visible, this is comparatively unimportant. For the same reason the complete separation of the shells, requires a tool of greater proportionate width, than with the Chinese ball; that the corner a, fig. 579, may quite complete the external faces of the more central shell, without b, cutting through those in the shell exterior to it.

Fig. 594.

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Fig. 595.

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Fig. 596.

Geometrical Solids Continued 400348

The methods of turning simple or compound solids with polyhedral envelopes, may be sufficiently gathered from preceding pages. In one instance, the compound solid turned within the sphere, fig. 569, is precisely the same turned within the cube, fig. 594; while figures of similar or different character may be produced within all the other regular solids, provided the number of points or other forms, corresponds either to that of the faces or the angloids of the solids. The central pins, employed to determine the depth of the faces, in figures 594 - 596, usually remain to form the apex of the cones, which are turned to a gage similar to fig. 571, and project beyond the faces of the external solid.

The production of one face, pin and cone, and hollowing the corresponding portion of the shell, may be concurrent and completed at all the faces, seriatim ; or, all the faces and pins may be first produced, and then returned to in regular order to turn every cone, and to hollow its portion of the shell. The combined filling pieces and stoppers, fig. 593, required as the points and faces are completed, are fitted to the apertures in the shell, pierced and exactly fitted to the cones, abutting against their base the central solid; affording support in the same manner as for the sphere, fig. 569. The edges of the apertures may be enriched by ornament, and the internal solid may also be contained within two or more shells, separated from one thicker envelope.