426 
d v d 2 , and d 6 are projections of the poles of three faces of 
the rhombic dodecahedron. 
Join G t d 5 , G 2 d 2 , and G 3 d x meeting in o 1 ' 3 o x is the projection 
of the pole of a face of the octahedron. 
(Fig. 43*, Plate IV *)— B 1? E 1? ¥ v Gr v H 4 , K v L v M p N p P 4 , 0, 
represent the poles of nearly all the known four-faced cubes 
lying in the arc of the zone d 5 G 3 ; B 2 , &c ., in G 3 d 2 ; B 3 , &c., in 
G x d 2 ) B 4 , &c., in G 2 d 5 ; B 5 , &c., in G 2 d 5 ; and B 6 , &c., in G x d 5 . 
Six poles of each four-faced cube in the octant at equal distances 
from G v G 2 , and G 3 . 
Pules for finding the position of B v E 4 , &c., will be given 
hereafter. 
1-^ij 1 4 ^ b 2 , e 2 , &c., I 2 ; and b 3 , e 3 , &c., 1 3 , 
three poles of each three-faced octahedron, lying at equal 
distances from o v in the arcs of zones represented respectively 
by o x d 53 o x d 23 and o x d v 
^i> e vfv 9v ^i> °v JPif ail d g x j b 2 , e 2 , &c., g 23 
b 3 , e 33 &c., q 3 , three poles of each twenty-four-faced trapezo- 
hedron, lying at equal distances from o 13 in arcs of zones 
represented by o x G 3 , o x G 13 and o 1 G 2 respectively. 
Lastly A 13 B v F 13 F v G v E v K v L v M v N v F v Q v B v S v 
-^ v ^1 > ^- 2 ) -^2’ G~ 2 ) A 3 , B 3 , &c ; , U 3 ; A 4) J5 4 , &c., Z7 4 ; 
A 5 , B 5 , &c., Z7g ; and A 6 , JB 6 , &c., Z7 6 , six poles of the six- faced 
octahedron ; the poles of each particular six-faced octahedron 
being similarly situated in each of the six triangles d 3 o x G 3 , 
d 2 °iC 3 , d 2 o 1 G 1) d^o-fi^ d 1 o 1 G 2 , and d 1 o 1 G 1 respectively. 
130. To find geometrically the position of any pole on the 
gnomic projection (fig. 43*, Plate IV.*). 
In (fig. 44*, Plate IV.*). — Let AG 3 , AG 2 , and AG 1 be three 
adjacent cubical axes, rectangular at A. 
Let AG 3 = 1. Take AN in AG t produced equal to n. 
AM in AG 2 produced equal to m. 
Join cyv, NM, MG 3 , G 3 G 2 , G 2 C v and G X C 3 . 
Then G 2 MN is the plane lmn 3 and G 1 G 3 G 2 is the plane of 
the gnomic projection. 
Through A draw AG perpendicular, G 3 M meeting G 2 G 3 in g, 
AH perpendicular G 3 N 3 cutting G 1 G 3 mh, and vUTperpendicular 
to G X G 23 cutting G 1 G 2 in It. 
li, g } and It are the projections on G 1 G 2 G 3 of H, G, and K. 
Join NGj MH 3 and C 3 K in the plane NMG 3 , meeting in F ; also 
join AF . Then, as in § 1 1 7, F is the pole of 1 m n 3 G of 1 m 00 , 
H of 1 00 n, and K of 00 1 — . 
m 
Therefore on the plane of projection, G X G 2 G 33 g is the pro- 
jection of the pole of lmoo, h of loo n, and k of 00 1 — ; 
m 
hC 2 of the line HM 3 JcG 3 of the line KG 3) gC x of the line GN. 
