66 
METHODS OF PETROGRAPHIC-MICROSCOPIC RESEARCH. 
even in it the length of a degree (both for radial polar angles and for tangen- 
tial azimuthal angles) near the horizon (margin of the projection) is nearly 
twice that of a degree near the zenith (center of the projection). In the 
orthographic projection the distortion is in the opposite sense, the degrees 
near the margin being too closely crowded. The result of such distortion 
is a variation in the relative accuracy of different parts of the projection 
plat, a millimeter polar distance representing i in one part of the projection 
plat and only 5 in another part, while on the sphere i has the same length 
of arc throughout. It would be obviously better for many purposes if this 
distortion could be reduced as far as possible. This has been attempted in 
the so-called equidistant projections by placing the eye at a point c (Fig. 39) 
intermediate between the stereographic point of view S and the infinitely 
distant orthographic point O. In de la Hire's projection of 1 701 , this point 
is located at 1.7071 (=i + V / times the radius below the center of the 
sphere, the result being that the polar distance for 45 is exactly half the 
radius of the projection plat. In Lowry's projection of 1825 the eye-point 
is 1.69 times the radius below the center of the sphere. The best average 
value for c may be obtained by transforming equation (ia) above to read 
c = 
x cos p 
sin p x 
(2) 
In this equation it is desired that the polar distance x in the projection be 
proportional to the polar angle p or 
x : i = p : 90 
substituting this value in (2), we have 
c = 
cos p 
go 
(3) 
sin p 
90 
from which the mean value of c can be 
found by determining by integration the 
area included by the curve from p = o to 
p = 9o: but this integration is so complex 
that a method of mechanical quadrature 
is simpler. From equation (3) the accom- 
panying values for c were calculated, from 
which the mean value of c was found 
by Euler's method of mechanical quad- 
rature to be c = i .69 1 7. With this value c 
the distances x for the different angles p 
were calculated and are listed in column 
5, Table 3. 
For the sake of comparison the corresponding values .t for the same polar 
angles p are listed (Table 3) for the gnomonic, the stereographic, the ortho- 
graphic, and the angle or equidistant zenithal projections. In this last pro- 
p 
c 
P 
c 

5 
759 
75'4 
50 
55 
.6966 
.6849 
10 
7498 
60 
.6720 
"5 
7470 
65 
.6380 
20 
.74^1 
70 
.6429 
25 
.7381 
75 
.6266 
3<> 
.7320 
Bo 
.6092 
35 
.7248 
85 
.5906 
40 
.7165 
90 
.5708 
45 
.7071 
