a re ee ee 
i 
_ In the first vertical, below H, 
On the Drawing of Figures of Crystals. 39 
If the eye be elevated above the horizontal plane, the lines PY’, 
NU’, HZ will be projected below GG. The lengths of these 
projections are in direct ratio to the lines projected. 
To obtain the values of PY’, NU’, HZ, we observe that 
PY’=cos5; NU'=cos (60°—95); HZ=cos (60°+ 9). 
Whence, since cos=1/(R? —sin?), 
cosd=/(1—sin? 6) = 5734; 
cos (60° —d)=4/(1—4 sin? 0)= 4/31; 
cos (60°-+45)=/(1—9 sin? 0)= / 35. 
From these equations, the following relations result: 
eNO MAAS AY YT: 
which, therefore, is also the ratio of the projections of these lines 
below GG consequent on an elevation of the eye at any angle «. 
1 
If the second projection of the semiaxis I (y/P), = 5 part of PM 
(the first projection), 
cot e=scotd=5s 1. 
For if P Y’ be made radius in the two triangles P Y’y' and PYM, 
we shall have Py’=tan ce, and PM=tan6. But Py’: PM: 1: s; 
consequently tants {tan Ort cs 
7. cot e=scoto. - 
In general it is most convenient to assume 2 as the value of s; 
then’ e—9° 500 lf s= 4/3; = 1101185". 
14. Projection of the axes.—The above demonstration affords 
a method of projecting the tetraxonal axes, which is similar to the 
method in the monometric system. We may assume r=3, s=2. 
1. Draw the lines 4.4, HH Fig. 7. 
at right angles with, and bisect- 7 
ing each other. Let HM=6, 
or HH=26. Divide HH into 
six parts by vertical lines. These 
lines including the left and right 
hand verticals may be numbered 
from one to six as in the figure. 
1 
lay off HS=~— 6, and from S 
