The regular trapezohedron may be sawn from the regular octangular prism, by means of two beds, one of them inclined in two directions. The first bed for the frusta of the two central pyramids, is inclined 21 degrees horizontally, or on the line B C, fig. 756. The second bed for the two exterior four-sided pyramids, is inclined 59 1/2 degrees horizontally on the line B C, fig. 756, and 22 1/2 degrees vertically, as at b, in the same group, in order to twist the prism on its axis, because the four terminal planes run on to the angle of the octagon.

* Mathematically, 10*. 48'. 44".

† Mathematically, 52o. 37'. 21".

The four planes of the terminal pyramid produce trapeziums, and which are increased, by trial, until they just equal the eight trapeziums formed by the partial obliteration of the central pyramidal faces. The second four-sided pyramid, which completes and releases the solid, is merely an inversion of the first.

The irregular or mineralogical trapezohedra, may be produced from the regular octangular prism, nearly in the manner just explained, by the employment of different angles, that are stated exactly in the annexed table, which shows the comparison of the three varieties of this solid selected for illustration.*

 Alternate angles of the solids. Beds for the central parts. Beds for the terminal parts. A. C. E. G. B. D. F. H. Hor.angles. Wedge. Hor. angles Vert. angles. Reg. Trapezohedron 135°. 0'. 135°. 0' 20°. 56'. none 59°. 38'. 22°. 30'. Irreg. --------- 126°. 52'. 143°. 8'. 24°. 6'. 8°. 8'. 54° 44'. 22°. 30'. --------- ---------------- 143°. 8'. 126°. 52'. 17°. 33'. 8°. 8'. 64°. 46'. 22°. 30'.

The table supposes the regular octangular prism to be in every case used, but to produce the irregular pyramid from the regular prism, requires the use of a wedge, as explained in page 773, and the angle of the wedge is half the difference between the two external angles of the prisms, which are simply the reverse one of the other. The wedge becomes unnecessary, if prisms are prepared, having the same irregular section that occurs in the second and third solids, and which is the preferable mode. If the lathe with revolving cutters and dividing plate is used for preparing the prisms, as hereafter recommended, instead of stopping the lathe at eight equal spaces, or taking 45° each time, the angles taken alternately, are the supplements to the two external angles of the prism, common to the second and third solids, namely 53°. 8'. and 30°. 52'., which together are equal to 90°.† When

* The Irregular trapezohedron, in another of its sections is a regular hexagon, as illustrated by the figure 772; 6ix of the trapeziums then con-stitute parts of the original prism, three trapeziums at an obtuse angle form the summit of the crystal, and three pairs of trapeziums are situated more acutely and intermediately, The trapezohedron might be therefore also worked from the hexagonal prism, by aid of two beds of the particular angles, one of them having a double inclination.

Fig. 772.

† The angles for the dividing plate are consecutively as follows:

 1 53°, 8'. 1 90°.
 3 143°, 8'. 4 180°.
 5 233°. 8'. 6 270°.
 7 323°. 8'. 8 360°.

Unless the lathe has an index with an adjusting screw, the 8' must in each case be neglected, but it is an admissible error.

the wedge is thus dispensed with, the vertical angle 22°. 30'., suit-able to the regular prism, becomes 18°. 26'. for the second, and 26°. 34'. for the third solid in the table, or half the supplements.

The order of proceeding given, in reference to producing the various solids with the circular saw, namely, first to saw the central parts of the solids, and then the terminal planes or pyramids, is in all cases advisable when only one or two solids of a kind are made, as the equality of the faces is then arrived at by two adjustments in place of four. The two central portions are simply inversions one of the other, and necessarily agree without trial; the central part thus produced, serves as the base from which to determine the two adjustments for the terminal parts.

As however, every step of this process depends on the primary accuracy of the prism, which serves as the means both of guiding and holding the pieces whilst under formation, it is desirable, as regards the more complicated polyhedra, that those who possess the lathe with revolving cutters, for ornamental turning, should make, or at any rate finish the prisms therewith, which will thence acquire an unexceptionable degree of accuracy. The trouble of preparing the wooden prisms, may be entirely saved, if metal prisms of the several sections, each with a conical hole to serve as a driving chuck, are prepared. The pieces of wood for the solids are then roughly turned, as cylinders with conical stems, which arc driven into the prisms for their attachment. The metal prisms may be used for an indefinite number of pieces; they save much trouble and uncertainty, and are especially desirable in the more complex polyhedra.

There are other and very different ways of making the geometrical and crystallographical solids. Sometimes the wood is prepared with the plane alone, into prisms of unequal sides and angles, so arranged, that two or four of the sides of the solid, may be parts of the surfaces of the original prism, and that some of the edges of the solids may fall on the remaining faces of the prism. The plane is then used subsequently to the saw machine, in perfecting and smoothing all the faces.

These modes do not admit of the same generalisation or facility of method as that described, which the author believes to be original, and that may be called the method of double pyramids; and which he was led to work out practically to the extent set forth,in order to show how much may be done by the saw-machine