This section is from the book "Turning And Mechanical Manipulation", by Charles Holtzapffel. Also available from Amazon: Turning and Mechanical Manipulation.

A much easier and more accurate way of cutting the second pyramid, is suggested by the author in figs. 757 and 758. The prism is in all cases to be left longer than the two pyramids, the first of which is cut as in fig. 756. Then leaving all matters as before, for pyramids of 4, 6, or 8 sides, simply to remove the parallel guide sideways, so as to change the position of 756 into 757, in order that the saw may enter the opposite side of the prism, at the base of the first pyramid, and proceed into the solid prism as far as its center. In a 4, 6, or 8-sided prism, the 4, 6, or 8 cuts release the double pyramid in 757, from its hollow bed, or inverted pyramid, or that which is sometimes termed, l>y mineralogists, its pseudo-morphous crystal. It is needful the saw should penetrate slightly beyond the apex, and the crystal will jump out of its bed when the last side is nearly cut through, leaving a trifling excess on the last side, just at the point; but if the inverted cuts are extended much beyond the apex, the model will be released before the last side is completed.

For double pyramids of 3, 5, or 7 sides, meeting base to base, as in fig. 752, the position of the saw in fig. 757, cannot be employed in cutting the second pyramid; because in a pyramid with uneven sides, the saw then would enter at one of the angles instead of at one of the faces of the first pyramid. Consequently the angular guide, fig. 756, is changed end for end, as in fig. 758, and all the sawing is done on the same side of the axis of the prism. The position fig. 758, might be used for all second pyramids, whether of odd or even sides, but for the latter the guide fig. 757, is more conveniently placed.

Sometimes, however, it is required that the face of one pyramid should meet the edge of the opposite, as in fig. 754, thus producing what is termed in mineralogy, a macled or twisted crystal. Macled double pyramids with 8, 5, or 7-sides, are cut by pursuing throughout the method prescribed for ordinary double pyramids with 4, 6, or 8 sides; namely, using the one guide, after the mode fig. 756 for the first, and after the mode fig. 757 for the second pyramid, and then with pyramids of uneven sides the required displacement is obtained.

Macled double pyramids, with 4, 6, or 8 sides, require the face B C, of the first guide, fig. 757, to be perpendicular as in the reduced figure a 758, and the face B C, 757, for the second pyramid, to be inclined 22 1/2, 30, or 45 degrees respectively, as at b, or half the supplement to the external angle of the respective polygons. For macled hexagonal pyramids, the side B C, may continue perpendicular, provided that in sawing the second pyramid, the edges, and not the faces, of the 6-sided prism are placed against B C, fig. 757.

Irregular prisms may be sawn into irregular pyramids, but certain corrections are sometimes required. Thus, the prism beneath fig. 759, which is more, and fig. 760, which is less than a regular hexagon, produce the irregular pyramids respectively annexed; the sides of each of which meet on one base line. In the first pyramid, fig. 759, the plain ridge is equal to the central piece added to the hexngon: in the second pyramid, fig. 760, the central face that corresponds to the narrow side of the hexagon, terminates below the extreme point. The six faces might in either case be made to converge exactly to one point, by employment of a second guide adapted to the irregular side.

Irregular pyramids, having as in fig. 763, equal sides, but unequal angles, produce pyramids, that converge exactly to a point.

Thus fig. 761 shows the result when the rhombic prism is cut into a pyramid, the bases of the sides also meet on one plane, and when the piece is released by cutting the inverted pyramid by the method shown in fig. 757, the solid that results is an irregular octahedron, the section of which is rhombic in both planes.

To produce an irregular octagonal pyramid from a regular octagonal prism, a wedge is placed beneath the prism, as in fig. 762, which now represents the guide; the point of the wedge is to the left, in cutting the sides 1, 3, 5, 7, of the octagon, and the point of the wedge is to the right, in cutting the sides 2, 4, 6, 8. By this twisting of the axis, the regular prism yields an irregular pyramid of the section shown at fig. 763, and the departure of the latter from the true polygon, is shown by the angular space, between the true polygon, and the vertical face in fig. 762, which space represents the piece removed in virtue of the subjacent wedge, the angle of the two being alike.

When the inverted irregular pyramid is similarly cut, the line of junction of the two is in one plane when the more obtuse edges of both pyramids meet; but the line of junction becomes zig-zag or macled, when the more obtuse angles of the one octagon meet the less obtuse of the other.

The same method punned with the 4 or 6-sided prisms produces similar results, subject, however, to certain displacements of the edges and points, the modes of correcting which will be sufficiently manifest to those who take up these matters practically.

It is now proposed to show how, by pursuing the methods of cutting various pyramids, the five regular solids, and many others, can be obtained with the saw-machine.

The tetrahedron, with 4 planes each an equilateral triangle, is cut from a regular triangular prism, inclined 19 1/2 degrees,* and it is best to cut it at the end of a long piece, as in fig. 756, and then to remove it by one cut of the saw at 90 degrees, which at any distance between the apex and base, produces the true tetrahedron.

The hexahedron or cube, with 6 planes each a square, is cut off from a square prism held at 90 degrees; the length of the piece removed, must necessarily be the same as that of the sides of the prism.

The regular hexahedron or cube, may be also viewed as two triangular pyramids, the faces of which are interposed or macled, or so placed, that the face of the one pyramid meets the angle of the opposite, as before explained in fig. 754. And pursuing this method, the cube may be sawn from a triangular prism by the positions figs. 756 and 757, provided the prism is inclined exactly 35 1/4 degrees to the saw.† The cube, when produced in this manner from the triangular prism, is however very small, as viewed diagonally, (and in which direction it is cut,) the cube appears as a hexagon, three angles of which touch the centers of the triangular prism. It is better to use the hexagonal prism, and to place its alternate sides, 1, 3, 5, successively upon the platform, both for the first and second processes, figs. 756 and 757; in which case the hexagonal outline of the cube, may be as large as the section of the hexagonal prism from which it is sawn.

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