397 
Hence we see that the ratios of the octahedral and rhombic 
axes of the inscribed four-faced cube to those of the circum- 
m __ l ;a llir._ 
o 
scribing cube are each equal to 
1 +m 
putting it under the form R= 
Calling this ratio Z?, and 
we see that for the cube 
m 
m=co, J?=l; and for the rhombic dodecahedron m = 1, and 
therefore R- 
Hence for the four-faced cube R varies from 1 to 
When m — f, R=^; m=f, R=%; m=f, R=f; 
Ws= b R=ii m=2, -R=f; m=|, i?= T V; 
m—f, R = -f ; m = 3, R= f; m = 4, ZZ=4; 
w = 5, .R=|; m=40, P=|f. 
61. To inscribe the twenty -four faced trapezohedron in the cube. 
(Fig. 31 , Plate IY.) Describe the square AC 1 D 1 G 2 = one-fourth 
the face of the cube 0-fi-Of)^ (fig. 27). Join AZZ r Produce 
D l G 1 to O v and G 2 A to Zb. Make C l 0 1 and AD 5 each =AD r 
Join Produce AG X to M v and take AM=m, AC 1 being 
1, and m any whole number or fraction greater than unity, 
m determines the particular species of the twenty-four-faced 
trapezohedron. 
Join G 2 M meeting AD 1 in d v In AD 5 take Acl~=Ad v 
Join d^M cutting A0 1 in o v Join 0 1 o 1 and 0 1 c? 1 . 
Then in (fig. 11, Plate II.), describe fig. 27, Plate IY., and 
take the eight points, o v o 2 , &c., o 8 , in the octahedral axes so that 
io; = fe (fig ' 11)5= 5 81 * Plafce IV -)- And tte 
twelve points d v d 2i &c., cl 12 , in fig. 11, Plate II., so that 
Ad x Ad 2 Ad x 
(fig. 31, Plate IY.), as described in § 36. 
AD 1 AD~AD 1 
Then joining the points 0, d 3 and o, as shown in (fig. 11, 
Plate II.) the twenty-four-faced trapezohedron will be inscribed 
in the cube. 
62. If (fig. 39, Plate IY.) we describe a triangle having one 
of its sides C f 1 o 1 = G 1 o 1 (fig. 31), another side Gft^ Gyd-^ (fig. 31), 
and its third side ofl^ofig (fig. 31) ; 
Then, on the other side of the base 0 1 o 1 (fig. 39), describe 
the triangle C 1 cl 2 o 1 similar and equal to the triangle G 1 d 1 o v 
0 1 d 1 Oi<Z 2 will represent on a plane surface a face of the 
twenty-four-faced trapezohedron, and 24 of these faces, 
formed into a net and folded together will make a solid 
twenty-four-faced trapezohedron, which can be inscribed with 
YOL. ii, 2 f 
