signs are contrary, give to their difference the sign of 
greater for niealanean Tadd a few examples. . 
a ia RBA Ae ae ch aan 
the first point of the second: quadran 
ba) erste, of the first point of the first 
- <quadrant . 2. se + wee 
_ Real error of the first point of the third 
MORON as lg. on ie aoe ot «5 9 em OD 
Half sum or half difference. . . . . 
mit t error of the dot under trial 
» 
: : Example 2. 
~ For the point 45° of the second quadrant. 
Real error of the first point of the quadrant + 
Real error of the last point of the quadrant — 
- Half difference... 2. oe 
once gente ges ey nta oe Pie 
+ Example 3. 
Point 88°.6, or last point, of the third quadrant. 
Real error of the point 84°.4 of the third 
uadranti!) SC. enesete Sui Lite Anan 2EO 
Real error of the point 2°.8 of the fourth 
‘ Marae MAE NE Wat er kyl) with ee 2.9 
lait ote.) SMMAR? dire setae carols be eee 
Apparent error of the dot. under trial. . — 4.0 
Real errGh ss SHUT. Ae Sea erne hey a> oahimea 1D: 
Example 4, 
Point 88°.6, or last, of the fourth quadrant. 
Real error of the point 84°.4 of the fourth 
SUE Quadrant M2 ja. sensri el. roel. mi SHE 
Real error of the point 2°.8 of the first qua- 
SAE TVS Pe eR Pea TOR 
Half sum . anime omer See oe 
Apparent error of the dot under trial... + 9.5 
meer: ARAL OD Os Ble & Geir Or 
'~ Itis convenient, in the formation of the table. of real 
errors, that they should be inserted in the order of the 
numbering of the degrees on their respective quadrants ; 
although their computation a ae in 
the order in which the examination was ied on, or 
according to the arrangement in the table of apparent 
errors, ‘The first dot of the first quadrant having been 
‘assumed to be in its true place, the first of the third 
“quadrant will err by just the difference found-b 
¢ examination ; therefore these errors are alike in 
‘tables. The real error of the first dot of the second 
ant comes out in the first example ; that of the 
was found in like manner, and completes the 
. first line. It is convenient to put the error of the di- 
vision 90° of each quadrant at the bottom of each co- 
lumn, although it is the same as the point 0° on the 
following quadrant. The line 45° is next filled up; 
the second example shows this; but there is no occa~ 
sion to dwell longer upon this explanation ; for every 
gone, who is at all fit for such pursuits, will think what, 
heen been said fully sufficient. for his purpose. 
4 wever, I will just mention, that there can be no 
danger, in the formation of this table, of taking from a 
wrong line the real errors which are to be the criterion 
GRADUATION, 
875 
for finding that of the one tnder trial, because they 
are in the next line to it, the others, which intervene in 
the full table, not being yet inserted, The last course 
of all is, however, an exception ; for, as the examining 
microscopes could not be brought near enough to bisect 
the angle 2° 48’ 45”, recourse was had to that quantity 
and its half ; on which account the examination is pro- 
secuted by using errors at two lines distance, as is 
shown in the two last examples. 
When the table of real errors is constructed, the other 
table, although it is of no farther use, should not be 
thrown away ; for if any material mistake has been 
committed, it will be discovered as the operation of di- 
viding is carried on, and in this case the table of appa- 
rent errors must be had recourse to ; indeed not a figure 
should be destroyed until the work is done. * 
ing the angular value of the numbers in these 
tables, it may be worth mentioning that it is not of the 
least importance, 100 of ther being comprised in one 
revolution of the micrometer screw ; and, in the instance 
before me; 5.6 of them made no more than a second. 
It is not pretended that one of these parts was seen 
Original 
Graduation. 
All the 
computa- 
tions should 
be preser~ 
ved. 
beyond a doubt, being scarcely -5%.-5 of an inch, much ~ 
less the tenths, as exhibited in the tables ; but as they 
were visible upon the micrometer heads, it was judged 
best to take them into the account, 
Having now completed the two first sections of my 
method of dividing ; namely, the first, which consists 
of making 256 small round’ dots; and the second, in 
finding the errors of these dots, and forming them into 
a table; I come now to the third and last part, which 
consists in using the erroneous dots in comparison with 
the tabulated errors, so as ultimately to make from them 
the true divisions. 
It will here be necessary to complete the description 
of the remaining part of the apparatus, And, first, a 
little instrument which I denominate a subdividing sec- 
tor presents itself to notice. From all that has hitherto 
been said, it must have been supposed, that the roller 
itself will point out, upon the limb of the instrument 
to be divided, spaces corresponding to others previously 
divided upon itself, as was done in setting off the 256 
points: but, to obviate the difficulty of dividing the roller 
with sufficient exactness, recourse was had to this sector ; 
which also serves the equally important p of re- 
ducing the bisectional points.to the usual division of the 
circle. This sector is represented of half its dimensions 
by Fig. 5, Plate CCLX XXIII. It is formed of thin 
brass, and centered upon the axis at A, in contact with 
the UPR surface of the roller: it is capable of being 
moved round by hand ;. but, by. its. friction upon the 
axis, and its pressure upon the roller, it is su ciently 
prevented. from. being disturbed by accident. An in- 
ternal frame BB, to which the are CC is attached, moves 
freely, in the outer one, and by a spring D.is pushed 
outwards, while the screw E, the pomt of which touch- 
es the frame B, confines. the are to. its proper radius. 
The are of this sector is of about four times greater ra- 
dius than the roller, and upon it are divided the spaces 
which must be transferred to the instrument as repre- 
sented on a magnified scale by Fig. 4. Now, the angle 
of one of the spaces of the circle will. be measured. by 
sixteen times its angular value upon the sectorial arc, or 
22° 30’; but this does not represent. any number of 
equal parts upon the instrument, the subdivisions of 
which are to be 5’ each; for attra 
: ——isexactly 163, 
therefore so many divisions are exactly equal to a mean 
* This is a very useful hint, applicable on many occasions. 4 
. 
True divi- 
sions to be 
made from 
the errone- 
ous dots. 
Subdividing 
sector de- 
scribed, 
PruatEe 
CCLXXXIIT. 
Fig. 5. 
