Mr. Troughton on dividing Instruments. 121 
of distinguishing the + and — errors, the heads, as mentioned 
before, are numbered backwards and forwards to fifty. One 
of the microscopes remaining as before, remove the other to 
a position at right angles ; and, considering for the present 
both the former dots to be true, examine the others by them ; 
i. e. as before, try by the micrometer how many divisions of 
the head the greater half of the semi-circle exceeds the less, 
and note half the quantity 4- or — > as before, and do the same 
for the other semi-circle. One of the micrometers must now 
be set at an angle of 45 0 with the other, and the half diffe- 
rences of the two parts of each of the four quadrants registered 
with their respective signs. When the circle is a vertical one, 
as in the present instance, it is much the best to proceed so 
far in the examination with it in that position, for fear of any 
general bending or spring of the figure ; but, for the exami- 
nation of smaller arcs than 45 0 , it will be perfectly safe, and 
more convenient, to have it horizontal ; because the dividing 
apparatus will then carry the micrometers, several perfora- 
tions being made in the plate B for the limb to be seen through 
at proper intervals. The micrometers must now be placed 
at a distance of 22 0 30', and the half differences of the parts 
of all the arcs of 45 0 measured and noted as before ; thus de- 
scending by bisections to 11 0 15', 5 0 37' 30'', and 2 0 48' 45". 
Half this last quantity is too small to allow the micrometers 
to be brought near enough ; but it will have the desired ef- 
fect, if they are placed at that quantity and its half, i. e. 
4 0 13' 7", 5; in which case the examination, instead of being 
made at the next, will take place at the next division but one, 
to that which is the subject of trial. During the whole of the 
time that the examination is made, ail the dots, except the 
MDCCCIX. R 
