874 GRADUATION 
Original ig not'to be expected, or wished ; but it is of some im- 
Gradastion, portance, that they shouldbe all of the same size, con- 
’ centric, small, and round. They should occupy a po- 
sition very near the extreme border of the circle, as well 
to give them the greatest radius possible, as that there 
should be room for the stationary microscope and the 
other mechanism, which will be described hereafter. 
It must be noticed, that there is a clamp and adjust- 
ing screw attached to the main plate of Fig. 1; but, as 
it differs in no respect from the usual contrivances for 
quick and slow motion, it has been judged unnecessary 
to incumber the drawing with it. Y x 
Method of Now the roller having been adjusted, with one mt- 
making —_ eroscope H upon its proper dot on the circle, and the 
the dots. other microscope at the first division on the roller ; 
lace the apparatus of Fig. 3. so that the dotting point 
may vet directly over the place which is designed 
for the beginning of the divisions. In this position of 
things, let the feeler X be pressed down, until its lower 
end comes into contact with the circle ; this will carry 
down the point, and make the first impression, or pri- 
‘mary dot, upon the circle ; unclamp the apparatus, and 
carry it forwards by hand, until another division of the 
roller comes near the wire of the microscope; then 
clamp it, and with the screw motion make the coinci- 
dence complete ; where again press upon the feeler for 
the second dot ; proceed in this manner until the whole 
round is completed. ’ 
ac From these 256 erroneous divisions, by a certain 
She crore By Course of examination, and by computation, to ascertain 
the dots. | their.absolute and individual errors, and to form these 
errors into convenient tables, is the next part of the pro- 
cess, and makes a very important branch of my method 
of dividing. 
The apparatus must now be taken off, and the 
circle mounted in the same manner as it will be in the 
observatory. The two microscopes, which have divi- 
ded heads, must also be firmly fixed to the support of 
the instrument, on opposite sides, and their’ wires 
brought to bisect the first dot, and the one which should 
be 180° distant. Now, the microscopes remaining 
fixed, turn the circle half round, or until the first micro- 
scope coincides with the opposite dot ; and, if the other 
microscope be exactly at the other dot, it is obvious that 
these dots are 180° apart, or in the true diameter of the 
circle ; and if they disagree, it is obvious that half the 
permet by which they disagree, as measured by the 
ivisions of the micrometer head, is the error of the op- 
posite division ; for the quantity measured is that by 
which the greater portion of the circle exceeds the less. 
It is convenient to note these errors ++ or —, as the 
dots are found too forward or too backward, according 
to the numbering of the degrees ; and for the purpose 
of distinguishing the + bo -— errors, the heads, as 
mentioned before, are numbered backwards and for- 
wards to fifty. One of the microscopes ehenqe as. 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 other by them ; ¢. e. as before, 
by the micrometer how many divisions of the head the 
eater half of the semicircle exceeds the less, and note 
alf the quantity + or —, as before, and do the same 
for the other cemicircle. One of the micrometers must 
now be set at an angle of 45° with the other, and the 
half differences 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, 
Method of 
_ _ ® Tf 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 interfere with another, and therefore 1 examine the whole before I begin to compute. . 
_ nation, are for the present supposed to be in their true. 
less concisely. If the real errors of the preceding an 
it is much the best to proceed so far-in the examination 0. 
with it in that position, for fear of any general bending G™ 
or. spring of the figure; but, for the examination of 
smaller arcs than 45°, it will be perfectly safe, und more 
convenient, to have it horizontal ; because the dividing 
apparatus will then carry the micrometers, several per- 
forations 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° 30’, and the 
half differences of the parts of all the ares of 45° measu- 
red and noted as before ; thus descending by bisections — 
to 11° 15’, 5° 37’ 30", and 2° 48’ 45". Half this last 
quantity is too small to allow the micrometers to be 
brought near enough; but it will have the desired éf 
fect, if they are placed at that quantity and its half, 
7. e. 4° 18’ 7”.5; in which case the examination, in« 
stead 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 examina- 
tion is made, all the dots, except the one under exami~ 
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 exami-. 
nation exceeds the other, and carefully to tabulate half 
of their difference. : 3 
I will suppose that every one, who attempts to divide Thei 
a large astronomical instrument, will have it ngcenly me 
first. Dividing is a most delicate operation, and every. jefore 
coarser one should precede it. Besides, its being num- divide 
bered is particularly useful to distinguish one dot from 
another; thus, in the two annexed tables of errors, (see 
p. 380, 381.) 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 renders 
ed nearly impossible. i 
The foregoing examination furnishes materials for 
the construction of the table of half differences, or ap- appa 
parent errors.* The first line of this table consists of *™% 
two varieties ; ¢. e. the micrometers were at 180° dis- — 
tance for obtaining-the numbers which fill the columns 
of the first and third quadrant ; and at 90° for those of 
the second and fourth quadrant. The third variety 
makes one line, and was. obtained with a distance of 
45°; the fourth consists of two lines, with a distance of 
22° 30': the fifth of four lines, with a distance of 11% 
15’: the sixth of eight lines, with a distance of 5° 37/ 
30”: the seventh of sixteen lines, with a distance of 
2° 48’ 45”: and the eighth and last variety, being the 
remainder of the table, consist of thirty-two lines, and 
was obtained with a distance of 4° 13’ 7.5. © My 
The table of apparent errors, or half differences, just Tap) 
explained, furnishes data for computing the table of er 
real errors. The rule is this: let a be the real error of — 
the preceding dot, and 6 that of the following one, 
and ¢ the apparent error, taken from the table of half 
differences, of the dot under investigation; then is 
a+b 
2 
sion may not be so generally understood by workmen — 
as I wish, it may be necessary to say the same —s 
+ec= its real error. But, as this simple expreg- 
following dots are both +, or both —, take half their 
sum, and prefix thereto the common sign ; but if one 
of them is ++ and the other —, take half their difference, 
