Coast Survey of the United States. 235 
means, or more generally, and much more easily, by compensa- 
tion. 'The methods thus decided upon will determine the num- 
ber, as well as the form and combination, of the observations 
which are required to give the greatest probability in the re- 
sults; and these methods must be the constant rule for all ob- 
servations.” 
That portion of geomorphy which comprises the measure- 
ment of terrestrial angles of the first order, demands the nicest 
exactitude, because it embraces the consideration of the earth’s 
figure, and because the celestial observations of a geodetic survey 
are directly connected with it. That observations should be nu- 
merous, as one mode of compensation, is evident. 
Another question arises as to the best circumstances in observ- 
ing, and the number necessary to secure a very small probable 
error. ‘This question the superintendent has now resolved, by 
an application of the method of least squares for the first time on 
the survey. Its general and successful use by Prof. Lloyd, in 
the magnetic survey of Great Britain, may be seen in the eighth 
Report of the British Association. ‘There is reason to think that 
the method adopted by some engineers of selecting for observa- 
tion those times only which appear the most favorable in all their 
minutest circumstances, does not lead to the best results. Swan- 
derg in. Lapland may be here particularly referred to, whose ob- 
servations, very few in proportion to the time occupied, do not 
appear to have commanded the highest confidence. Now the 
mode introduced by Dr. Bache has been to determine, by repeat- 
ed trials, and by a rigid application of the method of least squares, 
- what is the number of observations which, taken under cireum- 
stances such as are ordinarily favorable, will reduce the probable 
error within the limits of deviation of the instrument, and of the 
observer. ‘That number, thus established, becomes the rule of 
conduct. ‘The application of this method has increased not only 
the economy, (a very important consideration, for economy is 
progress,) but the rapidity of the work in a striking manner, as 
may be seen from a comparison of the field work of last year 
with that of any previous year; and, what is most to be regarded, 
‘it secures a greater accuracy in the compensation of errors. Of 
six triangles the greatest difference from 180°, after allowing for 
the spherical excess, was 0’:6 of a second of space, that is 0/2 
to an angle, and from this the difference descended to nothing. 
