212 
PRESTON. 
dependency, the error from the angular measures varies with 
the particular values assumed by the trigonometrical func¬ 
tions in the intervening triangles. To specify actual figures 
obtained in practice it may be said that the mean error of a 
measured angle is 1".07, and that the probable error of an 
average side of 26,600 meters is 0 m .335, giving an error of 
tttow part in the junction line between the two bases 200 
miles apart. 
The probable error of a side increases as the square root 
of the number of triangles, and we find the following rela¬ 
tions : 
At 100 miles from the base.. .probable error.. 
At 120 miles from the base.probable error.. 
At 150 miles from the base.probable error.. 
so that we may assume 240 miles as the limit of the distance 
between bases where an accuracy of tu woo part is desired. 
When one base is brought to another by calculation, there 
will in general be four discordant elements, requiring four 
conditional equations for their complete reconciliation; these 
are length, azimuth, latitude, and longitude. 
As the base lines are susceptible of measurement to a far 
higher degree of precision than can be maintained in our 
best schemes of triangulation, so also is the triangulation 
capable of fixing points on the earth’s surface more accu¬ 
rately than astronomical observations. 
On the transcontinental arc we have about 80 astronomical 
latitudes, 40 telegraph longitudes, of which more than 20 are 
primary, and 60 azimuths. Many of these stations are above 
10,000 feet elevation—a few are above 14,000 feet—and all 
have been determined with especial reference to the demands 
of coordinate parts of the work in that particular locality. 
Parenthetically it is proper to speak of the precision of the 
arc, taken as a whole, and of its expense. Taking into ac¬ 
count all the sources of inaccuracy, it may be assumed that 
the distance from the Atlantic to the Pacific along the thirty- 
ninth parallel of latitude is known within about 100 feet, 
