no Mr. Troughton on dividing Instruments. 
which, being bisected, gave 30° more to complete the total arc, 
A second order of bisections gave points at 13 0 distance from 
each other; but that which denoted 75 0 was most useful. 
Now, from the known length of the radius, as measured upon 
the scale, the length of the chord of io° 20' was computed, 
taken off from the scale, and protracted from 73 0 forwards ; 
and the chord of 4° 40', being ascertained in the same manner, 
was set off from 90° backwards, meeting the chord of io° 20* 
in the continually bisectional arc of 8 ° 20'. This point being 
found, the work was carried on by bisections, and the chords, 
as they became small enough, were set off beyond this point 
to supply the remainder of the quadrantal arc. My brother, 
whom I mentioned before, from mere want of a scale of equal 
parts upon which he could rely, contrived the means of divid- 
ing bisectionally without one. His method I will briefly state 
as follows, in the manner which it would apply to dividing a 
mural quadrant. The arcs of 6o° and 30° give the total arc as 
before; and let the last arc of 30° be bisected, also the last 
arc of 15®, and again the last arc of 7 0 30' : The two marks 
next 90° will now be 82° 30' and 86° 1 5', consequently the 
point sought lies between them. Bisections will serve us no 
longer ; but if we divide this space equally into three parts, 
the most forward of the two intermediate marks will give us 
85°, and if we divide the portion of the arc between this mark 
and 86° 13' also into three, the most backward of the two 
marks will denote 85° 30k Lastly, if we divide any one of 
these last spaces into five, and set off one of these fifth parts 
backwards from 83° 30', we shall have the desired point at 
1024 divisions upon the arc from o°. All the rest of the divi- 
sions which have been made in this operation, which I have 
