400 
When m=oo and — ~0, & = 1 and i? 2 =l, and the twenty- 
712 / 
four-faced trapezohedron becomes the cube. 
Hence, referring to (fig. 11, Plate II.) we see that the twenty - 
four-faced trapezohedron is a variable form of an infinite 
number of species, varying from the octahedron as one limit 
to the cube as the other. 
If we represent this passage as in the instances of the three- 
faced octahedron, §41, and four-faced cube, § 59, we must 
raise the eight points o x , o 2 , &c., o 8 , from o 1 equal ± A0 1 in the 
octahedron (fig. 14) to O v fig. 8; at the same time raising the 
points d 19 d 2 , &c., d 19 along the lines AB lf AB 2 , &c., from d v 
d 2 , &c. (fig. 14), equal one-half AB 1} to the point B v B 2 , &c. 
(fig. 8); taking care that the point cl shall have such a relation 
to o that two adjacent triangles on each side of Co are in the 
same plane. 
68. To inscribe the six-faced octahedron in the cute. 
(Fig. 35, Plate IY.) Describe the square ACfDfd 2 equal 
one-fourth of the square 0 1 0 i 0 8 0 5 (fig. 2/). Join AB r Pro- 
duce D 1 C 1 to O v and C 2 A to X> 5 , making Cfi-^ and AB 5 equal 
AD Join OJD b . Produce AC 1 to IT and N. Taking HC X =1, 
make AM=M and AN—n ; m being any whole number or 
fraction greater than unity, and n any whole number or fraction 
greater than on. 
Join C 2 M, cutting AB 1 in d v Take AB~ = Ad v Join d 5 N, 
cutting AOi in o r Join C 1 o 1 . 
Then, in fig. 27, take 12 points, d v d 2 , &c., d 12f in AB V 
are each equal 
Ad 
AB 2 , &c., AB 12 , so that j — , 
Ah &c., — i* 
AD 2 ’ AD n 
JL & • 
to -A fig. 35, which can be easily done with proportional 
• 2 , &c., o s , in A0 1 , 
Ah, &c., As=A- 1 , fig. 35. 
AOJ AO q AOJ s 
AD x 
compasses. 
Also, in fig. 27, take eight points, o v 
A0 2J &c., A0 8 , so that 
JA- (~J 2 ^ u u 
Join the points C, cl, and o as in (fig. 10, Plate II.), and 
the six-faced octahedron inscribed in the cube will be shown 
in perspective. In a model showing the solid six-faced 
octahedron inscribed in a skeleton cube, each of the lines 
0 \o v 0 2 o 2 , &c., 0 8 o 8 , will be equal 0 1 o 1 fig. 35, and each of the 
lines DfLv B 2 d 2 , &c., B 12 d 12 , will be equal B x d v fig. 35. 
69. Fig. 36", Plate IY. Draw a triangle, C f 1 o 1 d 2 , such that 
-C x o v fig. 36,= ^, fig. 35; C x d 2 , fig. 3Q,-C x d v fig. 35; and 
o x d g, fig. 36, = o x d 5 , fig, 35, 
