136 MEASUREMENT OF BASE LINE, 
values in dealing with a base exceeding seven miles in length 
are so small that it is scarcely necessary to take account of 
them, and I only refer to them in this paper to show how the 
difference in the computed lengths of the tapes was disposed of. 
In order further to test the accuracy of the measurement of the 
whole line, a system of triangles was established by which the 
length of half the line was computed and compared with the 
measured length of the same portion. The differences between 
the computed and measured length of the half was found to be 
only -936 of an inch, which is a most satisfactory proof of the 
care exercised in the measurement as well as of the system 
adopted in carrying it out. 
The line was measured, as before stated, in grades from peg to 
peg in 100 feet lengths. These lengths represented the 
hypothenuse of a right-angled triangle of which the side next 
in length represented the actual horizontal length between the 
pegs. A reduction of the measured length to the horizontal 
had to be made and computed for each subdivision. The correc- 
tion, unless for the two terminal sections, was very small. The 
measured length being the hypothenuse, and the rise or fall 
between the pegs, as the case may be, being the smallest side of 
the triangle, it was easy to find the third side. For convenience 
of calculation the difference of the squares of these two sides 
was found by the well-known formula of multiplying the sum 
and difference of the sides, and then extracting the square root 
of the product. This gave the horizontal length. The total 
correction for reduction to the horizontal was 31°123 feet, of 
which 19°732 feet was for the Mt. Irving section, and 11°225 
feet for that of the Mt. Maria end. 
The base was computed by logarithms, so that each stage of 
the process, from the entry in the field notes to the final deter- 
minations, could be traced step by step. It was a more laborious 
process than the ordinary arithmetical way, but it has the — 
advantage of showing every detail of the calculation, and has 
