382 
Griginal It can hardly have escaped notice, on perusing the 
Graduation. hove account, how strictly Troughton has adhered to 
the maxim of Graham. Beginning with bisecting the cir- 
—— cle, he has, by his mode of examination, without a change, 
Trough- followed the principle through eight successive courses. 
ton’sme- The action of the roller is stepping, notwithstanding that 
thod of di- jt consists of the exactest species of contact, and closes 
viding- without a remainder; and so fully is he aware of this, that 
256 checks are employed. As far as the examination and 
computation is carried, Troughton’s method, for aught 
we know, may be as tedious as any other ; but, by the 
contrivance of combining his subdividing-sector with the 
roller, he throws the quinquisections and trisections al- 
together into a species of engine dividing, and thereby 
reduces the remaining part of the work to a labour lit- 
tle surpassing that of cutting the divisions. 
Mr Thomas After the above paper was published, several gentle- 
Jones suc- men were of opinion, that the celebrity of Troughton’s 
ceeds in di- | -aduation [was owing more to the hand and eye of the 
viding a g . o @ 7 
ircle by _attist, than to the method by which it was effected. 
Mr Trough- Troughton, it will be remembered, in one part at least 
ton’sme- of the paper, expresses himself of a contrary opinion ; 
thod. and, to shew that he was not mistaken, we insert the 
following letter, which was addressed to him by a risin, 
artist, who is not too conceited to profit by the skill an 
experience of a veteran in the art: 
Dear Sir, « Charing-Cross, 15th July, 1813. 
Letter from Having now performed the graduation of a 
Mx Jones circle by your method, I take the liberty of addressing to 
Monaghan you the following lines upon the subject. The method 
which was used at Mr Ramsden’s was, before your's ap- 
peared, considered as the best. I learned it in the course 
of instruction, and practised it with patience and perseve- 
rance, for it requires much of both. I may, therefore, 
be allowed to consider myself a competent judge of the 
two methods; and, without presuming more than be- 
comes me, give an opinion concerning them, - From 
various motives, I feel great pleasure in saying, that 
dividing by the eye is greatly preferable to the other me- 
thod; the saving in time is very great, and accuracy in 
the result certain. With these properties, which it 
possesses in the first degree, were I debarred from 
using it in future, I should return to the old method 
with the greatest reluctance. I do not hesitate to say, 
that I feel myself equal to the dividing of a circle with 
a degree of accuracy equal to any one except yourself, 
ger do I think I should be very far behind you) ; and 
shall solicit that practice, which alone can make me 
quite your equal in the art. With many thanks for 
your liberal communications upon this as well as other 
subjects, I am, Dear Sir, your most sincere and obliged 
‘o Mr Edward Troughton. Tuos. Jones. 
Mr Caven- A paper by the late Henry Cavendish, Esq. called 
dish’s pro-  “* An improvement in the manner of dividing Astrono- 
ny pall mical Instruments,” was published in the second part 
duating in. Of the Phil. Trans. for 1809. Mr Cavendish introduces 
struments, his improvement in the following words: 
« The great inconvenience and difficulty in the com- 
mon method of dividing, arises from the danger of bruis- 
ing of the divisions, by putting the point of the compass 
into them, and from the difficulty of placing that point 
mid-way between two scratches very near together, 
without its slipping towards one of them ; and it is this 
imperfection in the common process, which appears to 
have deterred Mr Troughton from using it, an there- 
by gave rise to the ingenious method of dividing de- 
scribed in the preceding part of this volume. This in- 
duced me to consider, whether the above-mentioned in- 
eonvenience could not be removed, by using a beam 
GRADUATION. 
compass with only ene point, and a microscope instead 
of the other; and I find that in the following manner © 
of proceeding, we have no need of ever setting the point ..~,, 
of the compass into a division, and consequently that gich’g 
the great objection to the old method of dividing is en- posed 
tirely done away.” ; thod of 
To thisend, MrCavendish proposes to have a frame for “uating 
supporting his beam compass, that shall rest upon the cir- * 
cle to be divided, and which, by bearing against the edge 
of the latter, may be turned round silliest altering its 
distance from the centre. The Saute of this frame is 
triangular, and nearly as large as the circle itself. One 
of the angles is placed outwards, and the opposite side 
forms a chord of about. 150° to the circle. One end of 
the beam compass is attached to the outer angle of the 
frame by a vertical joint, round which it may be turned 
from one side to the other at pleasure ; and the joint 
must be moveable in the direction of the radius, in or- 
der that the beam may be adjusted, so as to form the 
chords of different arcs. At the opposite end of the 
beam is a fixed point, wherewith faint arcs are to be cut 
across the line of division, and with which the divisions 
themselves are finally to be made. There should be a 
anne or slit cut out all along the beam, in which a dou- 
le microscope with cross wires is to slide, and which may 
be fixed at any required distance from the point. Two 
props, one to the right and the other to the left, are to 
support the end of the beam opposite to the joint in the 
two positions. 
ith this Fy emp Mr Cavendish shews how to 
perform the different operations required in n parm d 
an instrument, namely, to bisect, trisect, and quinqui- 
sect ; but does not follow the subdivision through their 
repeated courses. 
In the bisection of an arc, the distance between the 
point and the axis of the microscope, is to be taken as 
nearly as may be to the chord of half the are, and the 
joint adjusted, so that the cross wires and point may 
both at the same time coincide with the line of division. 
With the beam to the right, by moving the frame upon 
the circle, the wires of the microscope must be brought 
to coincide with the point that marks the left boun 
of the arc, and then a faint mark across the line of di- 
vision must be cut with the point. The beam is next 
turned to the left, and the point which bounds the are 
on the right, is by turning the frame to be set to the 
microscope, and another mark across the line made as 
before. It is evident, that if the opening between the 
microscope and point was exactly equal to the chord of 
half the arc, the two marks would coincide upon the 
line of division ; but if that opening was too t or 
too little, the marks would cut each other without or 
within the circle. But in neither of these cases would 
Mr Cavendish put ina dot. _ Instead of which, when in 
subsequent division these marks are to be used, as well 
as when the final strokes are cut, he would place the 
wire of the microscope by estimation in the middle be« 
tween them, where ee cut the line of division. 
In quinquisecting an arc, the opening being taken as 
near as possible, equal to the chord of a fifth part of it, 
bring the microscope to one extremity of the arc, and 
with the point make a mark across the line; bring the 
microscope to the mark just made, and with the point 
make another mark, &c. until four are put in; change 
the position of the beam, and from the other extremity 
of the arc set off four marks as before. If the chord in 
the above operation was not correctly taken, it is evi- 
dent that there will now be four double marks, and the 
spaces between them equal to each other, and five times 
greater than the error of the opening. The real point 
of quinguisection, reckoning from one end of the.arc or 
SS 
