The next eminent artists in original graduation are the brothers John and Edward Troughton. The former was the first to devise a means of graduating the quadrant by continual bisection without the aid of such a scale of equal parts as was used by Bird. His method was as follows: The radius of the quadrant laid off from 0° gave the point 60°. This arc bisected and the half laid off from 60° gave the point 90°. The arc between 60° and 90° bisected gave 75°; the arc between 75° and 90° bisected gave the point 82° 30', and the arc between 82° 30' and 90° bisected gave the point 86° 15'. Further, the arc between 82° 30' and 86° 15' trisected, and two-thirds of it taken beyond 82° 30', gave the point 85°, while the arc between 85° and 86° 15' also trisected, and one-third part laid off beyond 85°, gave the point 85° 25'. Lastly, the arc between 85° and 85° 25' being quinquesected, and four-fifths taken beyond 85°, gave 85° 20', which as before is = 5' × 2^10, and so can be finally divided by continual bisection.
The method of original graduation discovered by Edward Troughton is fully described in the _Philosophical Transactions_ for 1809, as employed by himself to divide a meridian circle of 4 ft. radius. The circle was first accurately turned both on its face and its inner and outer edges. A roller was next provided, of such diameter that it revolved 16 times on its own axis while made to roll once round the outer edge of the circle. This roller, made movable on pivots, was attached to a frame-work, which could be slid freely, yet tightly, along the circle, the roller meanwhile revolving, by means of frictional contact, on the outer edge. The roller was also, after having been properly adjusted as to size, divided as accurately as possible into 16 equal parts by lines parallel to its axis. While the frame carrying the roller was moved once round along the circle, the points of contact of the roller-divisions with the circle were accurately observed by two microscopes attached to the frame, one of which (which we shall call H) commanded the ring on the circle near its edge, which was to receive the divisions and the other viewed the roller-divisions. The points of contact thus ascertained were marked with faint dots, and the meridian circle thereby divided into 256 very nearly equal parts.