The cubes are first prepared as described on page 774, and their faces are rubbed smooth; in cutting their edges and angles, beds similar to fig. 779 are required. The latter may be made entirely with the saw; for example, the rectangular block is supported on the face A, and two incisions a b, each at 45 degrees, are made by means of the saw and protractor; then the piece being placed with B downwards, and with the face A, against the parallel rule, the perpendicular notch c, is sawn; the three cuts release a piece of wood, leaving a cubical matrix.

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Fig. 773, the cube with bevilled edges, requires that the edges of the cube should be parallel with the saw, and the guide is then placed, as in fig. 781; that is, before the protractor, which is set at zero, and s is the stop for the quantity each of the 12 edges is bevilled or truncated. Cubes with two bevils or planes on each edge, may be bevilled with the position 781, provided the guide is tilted up some 20 degrees, by fixing a wedge of 20 degrees beneath the guide, as dottrel in fig. 779; or otherwise by making. a similar bed, fig. 780, with the angles 25 and 65 instead of 45, which will make a rectangular notch, inclined 20 degrees, as in fig. 780, so that the wedge may be dispensed with.

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Fig. 774, the cube with three bevilled planes at each angle of the cube, (one angle only being shown,) is obtained with the position of fig. 781; but the protractor is then set about 10 degrees from 90, so as to cut off every edge of the cube by two cuts slightly inclined. The square face of the cube then becomes an octagon, if the facets meet as represented in dotted lines, or a dodecagon when the bevils do not meet. The bed, if also inclined vertically, as by the wedge in fig. 779, will duplicate the angular chamfers, and it is clear this elaboration may be carried systematically to any required extent.

Fig. 775, in which the angles of the cube are truncated on the diagonal, require that the bed, fig. 781, should be placed at 35 1/4 degrees,* and then the angles of the cube will be cut off nearly at 54 3/4 degrees to every plane, or at right angles to the diagonal, and this little facet, in like manner to the above, may be converted into three planes, somewhat after the manner of fig. 774, if so required.

When, as in fig. 776, the angles of the cube are so far obliterated, that the eight little triangular planes exactly meet, the cube is converted into the cubo-octahedron, a solid having six square faces and eight triangular faces, the whole of which are equilateral; one only of each is represented, to avoid confusion.

By pursuing the last method a little further, so that the triangular faces encroach upon each other, they first produce a little ridge intermediate to the neighbouring facets, and carried to the proper extent, convert each of the triangular faces, in fig. 770

* Mathematically, 35*. 15'. 52". the name angle as that employed to produce the cube from the regular prism with 3 or 0 sides, by six pyramidal cuts; and also the regular octahedron from the square prism.

into equilateral hexagons, in fig. 777; the six little square faces are all that remain of the original cube, and these squares are united by eight hexagons, all equilateral. The name of fig. 777 when perfected, is the ex-octahedron, and which implies that this solid may be also obtained from the regular octahedron, by obliterating its six points, which develope the six squares, and the hexagons are then consequently parts of the octahedron.

If, as in fig. 778, all the angles of the cube could be truncated by planes extending from angle to angle, the cube would descend to the octahedron. With the circular saw this is impracticable to the full extent, although some of the planes may be developed; but the mineralogist produces the octahedron from cubes of fluor spar, which splits diagonally from every point of the cube with great facility.

When the octahedron is produced by the cleavage of fluor, further reduction only makes a smaller octahedron, which form is thence described as the primary crystal of this mineral. In other minerals, the cube is the primary to the octahedron.

It is expected that enough has been said to show that, with a little contrivance in the carrying out of the methods advanced, a vast number of even the most complex models of geometrical and crystallographical solids, with plane surfaces, may be produced with comparative facility and great exactness, by the saw-machine; and the mechanical amateur will find it a somewhat fascinating study, especially if he be likewise interested in geometry or crystallography.

The circular saw should be rather stiff, and have fine teeth, as then the planes developed by the instrument will be tolerably smooth, and merely require to be rubbed slightly on a sheet of fine glass-paper, laid on a flat board or metallic surface; they are sometimes cleaned off on a wooden face wheel, on which powdered glass or flint is glued after the manner of glass-paper.

In concluding this section, the author begs to add that the whole of the various works described, subsequently to page 76G, may be executed by the amateur with the machine represented on that page, aided by the simple additions described. The remainder of the chapter refers to larger sawing machinery, principally used by manufacturers.