40 On the Drawing of Figures of Crystals. 
draw a line through MM to the fourth vertical. YY’ is the projection 
of the axis I. 
2. From Y draw a line to the sixth vertical and parallel with HH. 
From T' the extremity of this line, draw a line to N in the second 
vertical. Then from the point U in which T'N intersects the fifth 
vertical, draw a line through M to the second vertical; UU’ is the 
projection of the axis II. 
3. From R, where 7'N intersects the third vertical, draw RZ 
to the first vertical parallel with HH. Then from Z draw a line 
through M to the sixth vertical: this line ZZ is the projection of 
the axis III. 
4. For the vertical axis, we observe in fig. 6. that Z’/H=sin 
(30° — 5) =cos (60°+4) ; also MH =cos (30° — 6) =sin (60°+9). 
But sin (60° +4) : cos (60°+6)=3,/3 : 1=3 ; tan 30°. 
Consequently } MH: ZH::1: tan 30°, or ZH equals tan 
30°, in a triangle whose base is 1 MH. If therefore we lay off from 
N on the second vertical (fig. 7.) a line of any length and construct 
upon this line an equilateral triangle, one side NQ of this triangle will 
intersect the first vertical at a distance, HV, from H, corresponding 
to ZH in fig. 6.; for in the triangle NHV, the angle HNV is an 
angle of 30° and HN=: MH. MV is therefore the radius of the 
circle (fig. 6.) Make therefore MA=MA’=MV; AA’ is the 
vertical axis, and YY’, UU’, ZZ are the projected horizontal axes. 
The explanation of this. construction, is obvious from the preceding 
demonstration, and from the remarks under the monometric system. 
15. The vertical axis has been constructed equal to the hori- 
zontal axes. Its length in the several tetraxonal primaries may be 
laid off according to the method sufficiently explained. If lines be 
drawn through the extremities of the horizontal axes, parallel with 
the vertical axis, and the parts above and below be made equal to 
the vertical semiaxis, their extremities will be the vertices of the 
angles of a hexagonal prism, and by connecting them, we obtain 
the projection of this solid. A double hexagonal pyramid, the isos- 
celes dodecahedron, may be projected by connecting the extremities 
of the horizontal axes with each other and also uniting them with 
the extremities of the vertical axis. By drawing lines through the 
extremities of each horizontal axis, parallel to a line connecting the 
extremities of the other two axes, a plane hexagonal figure will be 
obtained which is the section of a hexagonal prism diagonal with 
the one above referred to; and by connecting the angles of this 
