386 
six of its solid angles, 0„ 0„ &c., C* touching the centres of 
the six faces of tie circumscribing cube. 
Fig 13 The Three-faced octahedron inscribed in the cube, 
six of 'its solid angles, C„ 0 2 , &c„ C„ touching the centres ot 
the six faces of the circumscribing cube. . 
Fig*. 14. The Octahedron inscribed in the cube, its six solid 
angles C l3 C 2 , &c., C 6 , touching* the centres of the six faces 
of the circumscribing cube. 
Cubical Axes. 
22. The lines formed by joining the opposite centres of the 
faces of the cube C 5 0 SJ and G % G± (fig. 27, Plate .), aie 
called the cubical axes of the cube. These three lines are 
equal to each other, and are perpendicular each to two opposite 
faces of the cube; they intersect in A, the centre of the 
cube In fig. 27 two other sets of axes are shown, tour U L U 7 , 
0A,‘ 0 3 0 5 , and OA, joining the opposite solid angles 
0 9 , &c., Og, of the cube ; six others, AAu JJ 2 V 12) & c ., 
J)J) M joining the opposite centres A, &c -t> As of the edges 
of the cube ; both sets of axes passing through A, the centre 
of the cube. The four axes 0 X 0 7 , &c., 0 4 0 8 , fig. 2/, Plate IV 
are evidently the four diagonals of the cube, and are represented 
fig. 9, fig. 10, &c.,to fig. 14, Plate II., by lines marked thus 
_I. 1.. The line BJ) m fig. 27, is parallel and equal to a 
line drawn from O l to 0 6 , and is therefore equal to a diagonal ot 
one of the faces of the cube. The 12 axes AAu AAs> , &c., 
DqD 8 , are therefore each equal to a diagonal of the face ot t e 
cube. These lines are thus represented > ng. 0, 
fig. 10 to fig. 14, Plate II. 
Octahedral Axes. 
23. If the equilateral triangle 0,0,0s, representing; one of 
the faces of the octahedron (tig. 33, Plate IV.) has its three 
sides bisected by d,, d z , and 0,(4, 0-/1,, and 0 3 d, be drawn 
meeting each other in the point o„ this point o, will repre- 
sent the centre of gravity of the triangle 0,0 2 0 s , and any ot 
the shorter lines do will be a third of the longer one, 0 d. 
The octahedron inscribed in the cube fig. 14, Plate II., has all 
its edges bisected by the points d lf d 2) &c., d i2 , and each equi- 
lateral triangle divided into six triangles by lines Cd meeting 
in o D o 2 , &c., o 8 , the centres of the eight faces of the octa- 
hedron. , i T) T) 
Tt will be Been in fig. 14 that the six axes, such as A ' 121 
