388 
is perpendicular td the three lines Oft,, 0 2 d 2 , and Oft v aiid 
consequently to the plane face Gftft, of the octahedron. 
Likewise by symmetry of construction each of the four axes 
Oft*], &c., Oft, , are respectively perpendicular to two opposite 
and parallel faces of the octahedron. 
27. From triangle A Oft, (fig. 27*) we have 
A 0*= 0 ± D 5 2 + AD*= 1 + 2 = 3. 
Therefore AO x —ft 3. 
In right-angled triangle G 1 Ad 5 ; Oft, 
Therefore 0 1 d,— ftl m 
But triangles Aoft, and ADft^^ are similar. 
= C^ + Ad^~ l + i=| 
Therefore 
Ao,_AD, 
3ST^ r 
Ad, 
AD . \Z"i . •> t y/S 
AO, v/jj 
Consequently J.o 1 =i+0 1 . 
Again by similar triangles Ad,o x and Oft, A. 
°A 
Adz 
d,A 
°A= 
{Ad, 
d,0, - 
28, Hence, referring to 
% —\Gftv 
Also Ad,=±AB,. 
(fig. 14, Plate II.), we see that 
when the octahedron is inscribed in the cube, the three cubical 
axes. Oft,, 0 2 0±, and Oft, join together the opposite solid 
angles of the octahedron. The four octahedral axes Oft 7 , 
Oft 8 , &c.. Oft,, pass through the centres of two opposite faces of 
the octahedron and are perpendicular to them. 
The points o 3 
o 2 , &c., 
being 1 one-third of the distance of 
the centre of the cube from the solid angles 0 Y , 0 2 , &c., of the 
circumscribing cube. 
Also that the six axes Dft n , &c., D,D S , joining the opposite 
centres of the edges of the cube, pass each through two 
opposite edges of the inscribed octahedron. The distance of 
the centre of the cube from the centre of the edge of the 
octahedron being half the distance of the centre of the edge 
of the cube from that point. 
29. Referring to fig. 27*, Plate IV.*, we have already shown, 
§ 25, that the angle Ad,G 1 = angle AOft,— 54° 44' 8", conse- 
quently, since GftJDft-^ is by construction a parallelogram, 
The angle 6^+6^= 54° 44' 8", and the 
35° 15' 52". 
angle O x AD,~ 
Hence the angle such as 0 1 A0 1 which any octahedral axis AO 
makes with any adjacent cubical axis AO is 54° 44' 8" ; and the 
angle such as 0 ± AD, which the octahedral axis OA makes with 
any adjacent axis AD, is 35° 15' 52". This latter axis is called 
a rhombic axis. 
