258 
PRESTON. 
This statement exhausts the three independent methods of 
determining the earth’s size and shape. But in a coun¬ 
try like the United States, where we are liable to have long 
lines running in every conceivable direction, a general method 
for all oblique lines finds frequent application, and, without 
going into details, it may be stated, first, that it is desirable to 
utilize all available data, and, second, to so combine them that 
consistent results will follow. If we take any line at random 
on the earth’s surface and determine its length, the geograph¬ 
ical positions of the extreme points, and the directions of 
each point from the other, we have six measured values, 
which combine in one case all the data used in the three in¬ 
dependent methods. This gives sufficient matter to com¬ 
pletely solve our problem, and, moreover, gives us one super¬ 
fluous condition, which must be made to harmonize with all 
the others. Here comes in the great work of geodesy, viz., to 
get from all the measures one consistent and satisfactory re¬ 
sult. Take the astronomical difference of longitude between 
the two points and their given latitudes. This data enables 
us to locate the points on a sphere of any size. Add to this 
the length of the line, and we are limited to a sphere of a 
given magnitude. Add the direction of one point from the 
other, and we are forced to change from a sphere to an el¬ 
lipsoid of revolution to satisfy the given data. Add still the 
reciprocal direction, and we have one condition more than is 
necessary for the complete solution of the problem, and the 
necessary discrepancies find adjustment by the method of least 
squares. In this reconciliation between measured values, 
which are always more or less subject to error, and theoreti¬ 
cal conditions, the expedient so often employed in scientific 
work is adopted. Approximate values for the earth’s dimen¬ 
sions are introduced and, by differentiation, relations are es¬ 
tablished between the corrections to the approximate values 
and the measured ones. It is true that certain suppositions 
are made that are not strictly true, but the error introduced 
through this procedure always falls within the degree of ac¬ 
curacy sought in the final result. For example, in a system 
