Mr. Troughton on dividing Instruments. 111 
called marks because they should be made as faint as possible, 
must be erased ; for my brother would not suffer a mark to 
remain upon the arc to interfere with his future bisections, 
Mr. Smeaton, in a paper to be more particularly noticed 
presently, justly remarks the want of a unity of principle in 
Mr. Bird's method ; for he proceeds partly on the ground of 
the protracted radius, and partly upon that of the computed 
chord ; which, as Smeaton observes, may or may not agree. 
Bird, without doubt, used the radius and its parts in order to 
secure an exact quadrant ; but Smeaton, treating exactness in 
the total arc as of little value to astronomy, would, in order to 
secure the more essential property of equality of division, reject 
the radius altogether, and proceed entirely upon the simple 
principle of the computed chord. The means pursued by my 
brother, to reach the point which terminates the great bisec- 
tional arc, is the only part in which it differs from Bird's me- 
thod ; and, I think it is without prejudice that I give it the 
preference. It is obvious that it is as well calculated to pro- 
cure equality of division, as the means suggested by Smeaton ; 
at the same time that it is equal to Bird's in securing the pre- 
cise measure of the total arc. It proceeds entirely upon the 
principle of the protracted chord of 6o° and its subdivision ; 
and the uncertainty, which is introduced into the work by 
the sparing use which is made of subdivision by 3 and 5, is, 
in my opinion, likely to be much exceeded by the errors of a 
divided scale,* and those of the hand and eye, in taking off 
the computed chords, and applying them to the arc of the 
instrument to be divided. 
* That Bird’s scale was not without considerable errors, will be shewn towards 
the end of this paper. 
