176 
PRESTON. 
respective' curve— e. g., the tan of 68° is 2.48, the tan of 85° 
is 11.43, etc. 
The secants which are necessary in finding the values of 
c and d are obtained from the curves S', S", S'", etc. To 
facilitate their multiplication by sin « and cos «, the curves 
are drawn so that the secants are vertical lines and count 
from the axis of abscissas. From 45° on they are found in 
a similar manner to the tangents, but below 45° the curve 
S is used, which gives three places with sufficient accuracy. 
Referring to the case before cited, where the tangent of an 
angle less than 45° is to be employed, let us suppose the case 
where d = 25°. The tangent by the construction already 
indicated would be found (plate 9) on the line j p at p', 
where j p' — 0.466. 
If we only desire two places, instead of reading the value 
from the lineup it may be read from the horizontal line at 
a distance of one unit from the axis of abscissas and we get 
0.47. This being on the same scale as the tangents beyond 
45°, the subsequent proceeding is in every way similar. 
Should three places be desirable, project p' to p". Then 
p" p'" = 10 X sin a tan 8 = 4.33, and the true value of 1.337 
sin a tan 8 required in the construction of a A will be found by 
projecting^" to the axis of ordinates, and thus determining 
the line p iy p y — 0.58, j ¥ being mad e=j o X 0.134. 
The same result by an analogous construction follows by 
taking both tan 8 and the factor 1.337 in their true propor¬ 
tion. This is not shown in the figure, to avoid a multiplicity 
of lines and letters. It may be added, however, that inas¬ 
much as the trigonometrical function by which tan 8 is multi¬ 
plied can never exceed unity, two places are sufficient for 
small values of 8, and especially in view of the fact that in 
the quantity ad we have the factor A, wliich is small, and 
in b B the quantity 15 appears in the denominator, both 
tending to reduce the number of necessary places. 
