568 Royal Astronomical Society. 
servations. The method in question, which is due to Gauss and 
Bessel, has only recently been introduced into geodesy. In all the 
geodetical measurements which were executed prior to the latter 
part of the last century, the errors of observation were of such mag- 
nitude that it was unnecessary to take account of the curvature of the 
_ earth, and the triangles were accordingly computed as if they had 
been on a plane surface. On this hypothesis the sum of the three 
angles of each triangle is 180°; and as the strict fulfilment of this 
condition is necessary for the computation of the triangles, the uni- 
versal practice was to apply arbitrary corrections to the observed 
angles, the observer usually assigning the largest correction to the 
angle which he supposed most likely to be erroneous. The large 
and excellent theodolite used by General Roy in his triangulation 
(begun in 1784) for connecting the observatory of Greenwich with 
the meridian of Paris, and the repeating circle of Borda, introduced 
about the same time on the Continent, gave the means of measuring 
terrestrial angles with far greater precision than had been obtained with 
the old quadrants, and the curvature of the earth became a necessary 
element in ascertaining the amount of the errors of observation. No 
alteration was, however, required on this account in the mode of 
correcting the angles, for as the spherical excess can be computed 
from approximate values of the sides to any required degree of ex- 
actness, the condition necessary for the computation of an individual 
triangle was still that the sum of the three horizontal angles should 
be equal to a known quantity; and in all the principal trigonome- 
trical operations of which accounts have been published (excepting 
Colonel Everest’s prolongation of the Indian arc of meridian, and 
some recent surveys in Germany), this condition (speaking generally) 
is the only one which has been attempted to be satisfied in computing 
the observations. But the observance of this single condition is not 
by any means sufficient to give the best representation of the whole 
of the observations: nor does it even suffice to give a determinate 
solution of the problem under consideration, for when the distance 
between two stations is computed through different series of triangles, 
each mode of computing leads to a different result. When the in- 
strument has been set up at every station, and the angles between the 
other stations visible from that point have been all observed, other 
geometrical relations are established which the corrected angles ought 
likewise to satisfy, and angles are obtained which cannot be made 
use of otherwise than in satisfying such relations. Now, it is mani- 
fest that any mode of computing the triangles in which any observed 
angle is not taken account of, or any geometrical relation among 
the parts of the figure not satisfied, or which does not allow to every 
single observation its due influence, cannot be regarded as satisfac- 
tory. In order to obtain the nearest representation of the whole of 
the observations, or the result which is affected by the smallest pro- 
bable error, it is necessary to solve the following problem, viz. to 
determine the corrections which must be applied to the observed 
angles in order that they may satisfy a/l the geometrical relations or 
equations of condition, and in order that the sum of the squares of 
the corrections may be an absolute minimum. A general solution 
