of the edge e f, on the edge f a over the apex of the crystal, is 112°, 10′, and that of f a on a b 123°, 55′, it will follow that the triangular planes a i b and c i c are not equilateral, but isoceles, triangles, of which the outer sides a b and e c are the longest, the two others being equal. Now, six isosceles triangles, similar to those of a i b and e i c, fig. 198, are not equal to the complement of a regular six-sided plane, fig. 197, as will be seen by fig. 199. The macle delineated by fig. 188, therefore cannot be a regular dodecahedron with triangular faces. By an attentive examination it will be generally found to exhibit only three or four sections of the prism similar to fig. 196: and although this circumstance is commonly attributed to interrupted crystallisation, that is not in fact the cause of its assuming that appearance.
In my collection there is a macle obligingly presented to me by Mrs. Lowry, of about half an inch in diameter, and almost perfect, which as it demonstrates that six sections of the prism, fig. 196, are not equal to the complement of the dodecahedron is highly interesting. It is represented by fig. 189, which shews, that instead of exhibiting, as in the preceding figure, equal and similar planes 1, 1, of equal portions of fig. 196, it has only 3, each of them, alternating with facets of another form, having between them a re-entering angle.
Let fig. 201 represent a close combination of three isosceles triangles, a b c, a c d and a d e, similar to those produced by the lines of section on fig. 198. Then let a f c d g represent one of those triangles, and one-half of each of the other two. By comparing the plane a f c d g with a f c d g of fig. 200, which is the plane that would be the base of each pyramid of the macle described by fig. 189, by a section between them, it will be seen that there is a perfect agreement between each; and it will also be seen