THE TRANSCONTINENTAL ARC. 
211 
They are mostly modern, but one of them, the “ Kent Island,” 
in Maryland, was used as early as 1844. It is not to be ex¬ 
pected that the degree of accuracy now required was pos¬ 
sible fifty years ago, and so we find this base with a degree of 
precision only about half that now attainable: 2 2 8 1 q o ' q ' of its 
length (5.4 miles), or 1.5 inches, may be given as the probable 
error of the assumed length. I shall later speak of the rela¬ 
tive accuracy of the different bases employed on this chain, 
but before passing to that it is desirable to examine the con¬ 
ditions that govern the number and location of base-lines in 
any system of triangulation. It may be assumed as a guid¬ 
ing principle, that check-lines should be measured just so 
often and at just such distances as will be necessary to con¬ 
trol the triangulation and keep it within the degree of ac¬ 
curacy sought. 
In the main triangulation east of the Mississippi river we 
may assume e-Toirofh part as the limit of error. From St. 
Louis to Colorado Springs 1 0 t> -< y is easily attained, while 
from the last-named point to the Pacific the error to be feared 
is only ^nnsWth part. 
If we then take the middle figure as the index of accuracy 
which it is proposed to attain, the line of reasoning would 
be as follows: 
From a great number of observations the mean error of a 
measured angle is ascertained. From the actual angles em¬ 
ployed and the length of the base-line the reciprocal of the 
weight of any assumed side is found. The square root of 
this last quantity multiplied by the first will give the mean 
error of the side in question, and the probable error will be 
two-thirds of this. Comparing the result with the length of 
the side, we get the limit of accuracy for the distance trav¬ 
ersed. Dividing by j/ 2 gives the effect of two base-lines, 
and hence we have the total error to be expected in the junc¬ 
tion line midway between the bases. It is understood that 
the accuracy of any given side is influenced both by the base 
and angle measurement; but while the error from the former 
is a function of the length of the side and transmits itself in- 
