the Divisions of astronomical Instruments. 239 
In order to ascertain the greatest possible error to which 
we are liable in the examination, let s denote in parts of a se- 
cond the greatest that can be committed in bisecting any point 
upon the limb ; then, since this error may occur at each end 
of the arc, it is evident that e in the expression deduced above 
( x 2 pe) will become 2s, and the expression itself 
x 4 pe. Hence the possible error will be z -^ 4 ,s = 2s at 180°; 
7 + * 4 £ == 3 s at 9 °° ; 7 + i 7 1 x4£ = 3.33 £ at6o 0 ; fx2s 
-f- — ^ - x 4 x 2s = 4s at 1 20 0 . The greatest error must there- 
fore lie betwixt 90 and 120°, and nearer to the extremity of 
the latter than of the former arc. At 105° it will be 5.506 ; 
at 111 0 it Vv ill be 5.50s — ^ . 1,5s x 4 x 2s = 9,70s ; and 
at 111 0 10' it will be 9, 70s — 1,04s (the excess of the error 
at m° above that at 11 2 0 ) -j- 3.33s = 12.86s, which will be 
found to be the greatest error betwixt 105 and 120°, and of 
course the greatest in the first semi-circle. In the other semi- 
circle, the process being the same, the possible errors must 
necessarily be the same at the same distances from the first 
point, reckoning the contrary way upon the circle. 
The magnitude of the quantity s will of course vary upon 
circles of the same radius, according to the excellence of the 
glass employed, and the accuracy of the examiner’s eye. It 
will seldom, however, exceed one second upon a circle, whose 
radius is one foot ; and in general it will not amount to so 
much. I find that I can read off', to a certainty, within less 
than three fourths of a second, and hence I conclude, that I 
could examine the divisions of my circle without being liable 
to a greater error than 9.63 seconds, and those of a circle of 
