of some of the principal fixed Stars , & c. 
Mr. Troughton, in most of his instruments, particularly if 
they were intended to move freely in azimuth, has preferred 
the other method. 
In this case, the plumb-line is attached to one of the pillars 
which support the microscopes in the way above described ; 
and it has no reference to any fixed points or divisions on the 
limb of the circle, but only insures a similarity of position in 
the index, for each position of the instrument ; and, provided 
that the plumb-line apparatus was free from all danger of 
derangement, this would be sufficient. This verification may 
be rendered perhaps more intelligible, by considering that a 
circular instrument, in whatever manner its vertical axis be 
placed, indicates by a double observation, the angle which the 
object makes with the axis, round which the whole instrument 
has revolved in passing from one position to the 
other. For let Y*p be the axis, Tx the telescope x / 
in one position ; it is evident, that in turning the 
instrument half round, ty will then be the position t 
of the telescope : P# being equal to Py. The arc 
xy, which the telescope passes through to regain its former 
position, is the quantity really given by the instrument ; and 
if the axis P p be vertical, haif this quantity is the true zenith 
distance of the object. Now the intention of Mr. Troughton's 
verification is to insure a vertical position to the axis P p. 
For instruments which rest on moveable pillars, and turn 
freely in azimuth, this method is much to be preferred ; but 
it is not without a considerable defect : for, if by any derange- 
ment in the plumb-line apparatus, the error in colliination be 
changed, it cannot be restored with certainty to its former po- 
sition ; so that sometimes a very valuable series of observations 
