381 
There are 12 longer edges, such as O x C it CA, &c., and 
24 shorter, such as oflu oA, &c. The 12 longer edges 
are the edges of an octahedron. It may be formea 
by placing on every face of the octahedron a three-faced 
pyramid on a equilateral triangular base. The angles of 
these isosceles triangles differ in different species of the 
three-faced octahedron, within certain limits to be described 
hereafter 
The synonyms for this form are the pyramidal octahedron , 
trialnsoctahedron, trioctahedron , and galenoid. 
16. The Four-faced Cube (fig. 2, Plate I.) is bounded 
like the last by 24 equal faces, each being an isosceles triangle, 
such as C l0l o 4 (fig. 34, Plate IV.), but grouped so together as to 
form a solid having six solid angles, u x , Co, Y G 
each formed by the union of four of the largest angles of the 
isoscles triangles, and eight solid angles, o l} o 2 , & c v> °s ( »• b 
formed by the union of six of the equal angles of the isosceles 
triangles. This form has 24 shorter edges, such as C&, 
O x o 2 , &c., and 12 longer ones, such as o x o 4 , o x o 5 , &c. Lhe i l 
longer edges are those of a cube. 
It may be formed by placing on every face of the cube a 
four-faced pyramid on a square base. 
The angles of the isosceles triangles differ for each particular 
species of the four-faced cube. 7 . 
Synonyms. — Pyramidal cube, hexatetrahedron , tetrahis- 
hexahedron, and fluoride. , 
1 7. The Twenty-four-faced Trapezohedron (fag. 4, Plate 1.) 
is bounded by 24 equal faces, each face being a deltoid or 
trapezium, C^oA (fig. 39, Plate IV.) ; that is , a four-faced 
figure having two longer equal sides, C 1 d 1 and C^, and two 
shorter equal sides, o x d 2 , oA. These 24 equal trapeziums aie 
so grouped together as to form a solid having six solid angles, 
Ci, C 2 , &c., C 6 , formed by the union of the plane angles of 
four trapeziums, equal to d 1 C x d 2 ; eight solid angles, o 1} o 2 , &c., 
o 8 , formed by the union of the plane angles of three trapeziums, 
equal to d x oA ; and 12 solid angles, d 1} d 2 , &c., d 12 , formed b} 
the union of the plane angles of four trapeziums, equal n 
CA G v This form has 24 equal longer edges, such as CA, CA, 
and 24 shorter edges, such as o x d 1} o x d 2 , &c.. The angles of tie 
deltoids or trapeziums differ for each particular species Oj. the 
twenty -four -faced trapezium . 
Svnonyms. — Icositessarahedron , icositetraliedron , trapezo- 
hedron, and leucitoid. # _ , 
18. The Six-faced Octahedron (fig. 3, Plate I.) is bounded 
by 48 equal faces, each face being a scalene triangle, CpA 
(fig. 36, Plate IV.). These 24 triangular faces are so grouped 
together as to form a solid having six solid angles, C x , C 2 , cvC., 
