Original 
Gradwation. 
— 
’ Dividing by 
878 
until we ¢ome to cut the divisions ; and for this pur- 
pose we must have another line divided upon the sec- 
tor. ' For ,4,>th part of the circle being equal to 5’.4 
1° 24! 29".5 Ae 
of the usual angular measure a = 15} divi- 
sions; and just so many will be equivalent to one of 
the intervals of the circle. The value of one of the 
great divisions of the sector willbe 1° 26’ 24’, and that 
of the th parts, which are to be annexed to the right 
and left as before, will be 10’ 48”, therefore divisible 
‘by the engine. Should any astronomer choose to have 
-both graduations upon his instrument, the additional 
cost will be a mere trifle, provided both were done at 
the same time. , 
It must already have been anticipated, that dividing 
the eyeap- by the eye is equally applicable to straight lines as it is 
plicable to 
straight 
lines as well 
as cireles. 
Apparatus 
to circles. An apparatus for this purpose should con- 
sist of a bar of brass, three quarters of an inch thick, 
_and not less than three inches broad ; six feet may do 
very well for the length; it may be laid upon a deal 
plank strengthened by another plank screwed edge- 
wise on its lower surface. The bar should be planed 
for the pur- on both its edges and on its surface, with the greatest 
“pose. 
Method of 
using it. 
It might 
be used for .mend it; because, 
dividing 
circles, but 
not to be re- <P 
commended, Wheel, are sufficiently reduced b 
exactness ; and it will be better if it has a narrow slip 
.of silver, inlaid through its whole length, for receiv- 
ing the dots.. An apparatus nearly similar to the other 
should slide along its surface, carrying a roller, the cir- 
-cumference of which is 12.8inches, and turned a little co- 
~nical for the sake of adjustment. The roller may be di- 
vided into 32.parts, each of which, when transferred to 
the bar, will give intervals of 0.4 of an inch each: the 
angle of the subdividing sector should of course be 11° 
15’, and subdivided into four parts, which will divide 
the inch into tenths: the surface may also receive other 
dines, with subdivisions suited to the different purposes 
tor which it may be wanted. The revolutions of the 
roller and its #, parts must be dotted upon the bar ; 
taking care, by sizing the roller, to come as near the 
true standard measure as possible: when this is done, 
compare the extent of the greater bisectional number 
that is contained in the length, 7. e. 128 intervals or 
51.2 inches, with the standard measure ; noting the 
difference as indicated by the micrometer heads: the 
examination and construction of the table of errors may 
ithen be conducted just as was done for the circle. 
Being now ready for the performance of its work, 
the scale to be divided must be laid alongside of the 
bar, and the true divisions must be cut upon it by an 
appeal, as before, to the erroneous dots on the bar, cor- 
rected by a corresponding table of errors. The appa- 
ratus, remaining entire in the possession of the work- 
man, with its primitive dots, the table of errors, &c. is 
ready for dividing another standard, which will be pre- 
cisely similar to others that have been, or may be, di- 
vided from it. It may be considered, indeed, as a kind 
_of engine ;. and, .as it is not vitiated by the coarse ope- 
ration of racking with a screw, but performed by only 
‘looking atthe work, the method will command about 
three times the accuracy that can be derived from the 
“ustial straight-line dividing engine. Should it be ask- 
ed, if an engine thus appointed would succeed for di- 
viding circles? I answer, Yes; but I would not recom- 
x beyond a certain extent of radius, it 
4s hot necessary ; for the errors, which would be intro- 
duced into the work by the violence of racking a large 
: y the comparative short- 
ness of the radius of such instruments as we divide by 
that method: ayd, what is still more to the purpose, 
* This paper was written in June 1808. 
GRADUATION. 3 q 
‘instances, 
the dividing engine is four times more expeditious, and Oni 
rougit tieige better. I cannot quit the subject of © 
dividing straight lines without observing, that I never 
had my apparatus complete. The standard which { Const 
made for ir George Shuckburgh Evelyn in 1796, was tion 
done by a mere make-shift contrivance, upon the prin- S700 
ciple of dividing by the eye; how I succeeded, may ~~ 
be seen in Sir George’s papers on Weights and Mea- 
sures, in the Phil. Trans. for 1798. Imadeasecond, 
some years after, for Professor Pictet of Geneva, which 
became the subject of comparison with the new measure 
of France, before the National Institute ; and their re- 
port, drawn up by Mr Pictet, has been ably re-stated ~ 
and corrected by r Young, as boa wee in the Jour- 
‘nals of the Royal Institution. I made a third for the 
magistrates of Aberdeen, I notice the ped latter, tet 
cipally to give myself an o' uni sayi 
Sf thbwe thas scale #reke 40 be Sinaia cheetah ane 
withstanding they were divided at distant periods of 
time, and at different seasons of the year, they would 
be found to agree with each other as nearly as 
rent parts of the same scale agree. ie a 
I hope I may here be allowed to allude to an inad- tnad 
vertency which has been committed in the bec aa men- tency 
tioned above ; and which Sir George intended to have &- 5! 
corrected, had he lived te conclude his useful endea- ly 
vours to harmonise the discordant weights and mea- se 
sures of this coun The instruments which he has 
brought into comparison are, his own five feet standard __ 
measure and equatorial ; General Roy’s -two inch 
scale; the standard of Mr Aubert; and of the 
Royal Scart, The inadvertency is this: in his 
torial, and the standard of the Royal Society, he. 
charged the error of the most erroneous extent, when 
compared with the mean extent, alike to both divisions; 
i.e lie Hes supposed one of the divisions, which bound 
the erroneous extent, to be too much to the right, atid 
the other too much to the left, and by equal quan- 
tities. This is'certainly a Ur gpeint oh way of stating 
the errors of work ; and perhaps not unjustly so, where 
the worst part has been selected; but in the other three 
namely, in General Roy’s, Mr Aubert’s, and 
his own standard, he has charged the whole error of 
the most erroneous extent to one of the bounding lines. 
I was well confirmed in my high opinion of the ge- Gen 
reral accuracy of Bird’s dividing, when, last win- cur 
ter, * I measured the chords of many ares of the Green- 3 
wich quadrant. That instrument his indeed suffered, 
both from a change in its figure, and from the wearing The 
of its centre ; but the graduation, considering the time ¥!* 
when it was done, I found to be very | . Sir an 
George, in his paper upon the equatorial, (Phi/. Trans. 
for 1793,) after some compliments paid to the divider 
of his instrument, says, “ the late Mr John Bird seems 
to have admitted a probable discrepancy in the divi- 
sions of his eight feet quadrant, amounting to 3’’;” and 
he refers to Bird on the construction of the Greenwich _ 
quadrant. This quantity being three times as gréat as _ 
any errors that I met with, I was lately induced to in- 
quire how the matter stood) © Bird, in the paper refér- 
red to, says, “in dividing this instrument, I never met _ 
with an inequality that exceeded one second. I will 
suppose, that in the 90 arch this error lay toward the 
left hand, and in the 96 arch that it lay towards the 
right, it’ will cause a difference between thé two arches 
of two seconds ; and, if an error of oné secénd be’ al- 
lowed to the observer in reading off his observation, the 
whole amount is no more than three seconds, which is 
