122 Mr. Troughton on dividing Instruments. 
one under examination, are for the present supposed to be in 
their true places ; and the only thing in this most important 
part of the business, from first to last, is to ascertain with the 
utmost care, in divisions of the micrometer head, how much 
one of the parts of the interval under examination exceeds 
the other, and carefully to tabulate the half of their diffe- 
rence. 
I will suppose that every one, who attempts to divide a 
large astronomical instrument, will have it engraved first. 
Dividing is a most delicate operation, and every coarser one 
should precede it. Besides, its being numbered is particularly 
useful to distinguish one dot from another ; thus, in the two 
annexed tables of errors, the side columns give significant 
names to every dot, in terms of its value to the nearest tenth 
of a degree, and the mistaking of one for another is rendered 
nearly impossible. 
The foregoing examination furnishes materials for the con- 
struction of the table of half differences, or apparent errors.* 
The first line of this table consists of two varieties ; i. e. the 
micrometers were at i8o° distance for obtaining the numbers 
which fill the columns of the first and third quadrant ; and at 
qo°, for those of the second and fourth quadrant. The third va- 
riety makes one line, and was obtained with a distance of 45 0 : 
the fourth consists of two lines, with a distance of 22°3o': 
the fifth of four lines, with a distance of 11 0 15': the sixth of 
eight lines, with a distance of 5 0 37' 30" : the seventh of six- 
teen lines, with a distance of 2 0 48' 45" : and the eighth and 
* If the table of real errors be computed as the work of examination proceeds, there 
will be no occasion for this table at all ; but, I think it best not to let one part inter* 
fere with another, and therefore I examine the whole before I begin to compute. 
