Mr. Troughton on dividing Instruments. log 
Bird, who enjoyed the undisputed reputation of being the 
most accurate divider of the age in which he lived, was the 
first who contrived the means how to render the usual divi- 
sions of the quadrant bisectional ; which property, except his 
being unusually careful in avoiding the effects of unequal ex- 
pansion from change of temperature, chiefly distinguished his 
method from others who divided by hand. This desirable 
object he accomplished by the use which he made of a finely 
divided scale of equal parts. The thing aimed at was, to ob- 
tain a point upon the arc at the highest bisectional number of 
divisions from o, which in his eight feet quadrants was 1024, 
= 85° 20'. The extent of the beam-compasses, with which 
he traced the arc upon the limb of the instrument to be divi- 
ded, being set off upon that arc, gave the points o° and 60% 
broad list, and thereby shews that there is a yielding between them, to a considerable 
amount. The breadth of this list is not less than one fiftieth of an inch ; and it follows, 
that at least jz° of the circle’s edge must be in contact at the same time; that the 
two surfaces yield to each other in depth, by a quantity equal to the ver. sin. of half 
that arc, or vsVo of an inch ; and that the circle has always hold of the roller by 
nearly i° of the edge of the latter. Whoever has examined the surfaces of metals 
which have rolled against each other, must have observed that peculiar kind of in- 
dentation that always accompanies their action ; and there can be no doubt that the 
particles of a roller, and those of the surface on which it acts, which mutually indent 
each other, will, upon a second course begun from the same point, indent each other 
deeper: This is not, however, exactly the case in question; for, whatever of fitting 
might have taken place between the surf aces of our roller and circle in the first revo- 
lution of the former, one should imagine would be obliterated by the fifteen turns 
which it must repeat over fresh ground. Experience shows, however, as every one 
will find who tries the experiment with good work, that on coming round to the point 
of commencement, the roller has the disposition to regain its former track ; for, were 
this not the case, although the commensurate diameters were adjusted so exactly as; 
to be without sensible error in one course, yet a less error than that which is so 
would become visible, when repeated through many courses. 
