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each face of the cube, G v C 2 , &c., C 6 , by one of its four-faced 
solid angles; cuts each octahedral axis AO v A0 2 , &c., by 
o lf o 2 , &c., one of its three-faced solid angles, at a distance Ao 1 
the of AO v Also each semi-rhombic axis cuts the centre of 
the rhombic face, such as G 2 o 1 C 3 o 5 at cZ 5 , Ad 5 being 
To inscribe the three-faced Octahedron in the Gabe. 
35. (Fig. 29, Plate TV.) Describe the square GJDfG^A, having 
each of its sides equal to 0 1 D 1 , fig. 27. Draw the diagonals 
G 1 G 2 and D X A meeting in d v 
Produce D 1 G 1 and G 2 A to 0 1 and D 5 , make AD 5 and G 1 0 1 each 
equal to AD V Join OJD,. In AB b take Ad b =Ad v 
Produce AG ± to M. For distance AM see § 37. Join d b M, 
cutting A0 1 in o 1 . Then join G^. 
Then referring to (fig. 13, Plate II.), G 1 d 1 G 2 represents the 
edge of the three-faced octahedron, G 1 o 1 and o x d b the corre- 
sponding lines shown in perspective. 
36. To draw the three-faced octahedron inscribed in the 
cube (fig. 27, Plate IV.). 
Describe a square ; draw 0 4 0 3 at such an angle and 
such a length that none of the edges or axes of the cube may 
obscure each other. Then draw 0 1 0 2 , 0 5 0 6 ,and 0 8 0 7 parallel and 
equal to 0 4 0 3 . Join 0 3 0 2 , 0 2 0 6 , 0 6 0 7 , and 0 7 0 3 . Also join 0 4 0 7 , 
0 2 0 8 , 0 3 0 5 , and 0 4 0 6 meeting in A, the centre of the cube. These 
diagonals of the cube are the four octahedral axes of the cube. 
Bisect Of) 2 in D v 0 1 0 2 in B 2 , &c., 0 8 0 7 in _D 12 ; join BfD lv 
B 2 B 12 , _D s D 9 , D 4 .D 10 , _Z) 5 Z) 7 , and D 6 D 8 , all intersecting in A. 
These are the six rhombic axes of the cube. 
Lastly take G 1 the intersection of the diagonals of the face 
OiO 2 O 3 0 4 , G 2 that of the diagonals of the face 0 1 0 5 0 8 0 4 , &c. 
Join Gj'Cq, G 2 G 4 , and G 3 G 5 intersecting in A. These are the 
three cubical axes of the cube. 
Then take a pair of proportional compasses and set them so 
that Ao 1 (fig. 29, Plate IV.) be the distance between the shorter 
. legs, and AO x between the longer legs of the compass. 
Then in fig. 27, take the distance A0 1 with the longer legs 
and mark off Ao 1 with the shorter ; in the same way mark off 
the points o 2 , o s , &c., o 8 , on the other octahedral axes. 
Lastly (fig. 13, Plate II.) prick off from this construction of 
(fig. 27, Plate IV.) the points G lf G 2 , &c., G 6 ; D v JD 2 , &c., D 12 ; 
O v 0 9 , &c., 0 8 ; and o 4 , o 2 , &c., o 8 . Draw the same lines as in 
fig. 27. 
Join G-fJ 2 ) G 2 G 8 ) & c ., G-^0-^ } G 2 0-^ } F 3 o 4> F 4 o 4 , F 2 o 4 , Lg 0 4 , &c. 
Then d 13 d 2 , &c., will be the points where the rhombic axes 
bisect the edges GfJ 2) GfG 8 , &c. Join with dotted lines d 1 o l , 
d 2 o v &c. ; then (fig. 13, Plate II.) will represent in perspective 
the three-faced octahedron inscribed in the cube. 
