meter, and the half of that quantity is noted in a table + or —, ac-
cording as its position is found forward or backward in the intended
order of the future divisions. The four quadrants are next examined
in the same manner, by removing the micrometer microscope to a
position 90° distant. Half the observed error is again the real dif-
ference of the two portions of semicircle compared; and since the
sum of the quadrants (though not accurately 180°) is known by the
previous examination, the quantities themselves are known, and. the
real error of each quadrant is then noted. In a similar manner the
succeeding bisectional points, at the distances from each other of 45°,
22° 30’, 11° 15', 5° 37’ 30", 2° 48' 45", and 1° 24' 22%", are suc-
cessively examined; and the real errors of the several dots. from their
true places, are computed and arranged in a table, so that by means
of the dots themselves, together with their tabulated errors, the true
places for the future divisions may be correctly known.
For the purpose of laying off these ultimate divisions, the circle is again placed in a horizontal position, and the roller is again applied to it. But as it would not be easy to divide the roller itself with sufficient exactness, a sector is added to the apparatus, having its radius four times that of the roller. This sector being fitted tight on the axis of the roller, moves with an angular velocity, which is six- teen times that of the circle; so that one of the former divisions of the circle is measured by an are upon the sector of 22° 30’. But since the ultimate divisions are intended to be 5’ each, this sectorial arc must be divided into spaces of 80' each; and of these spaces 16% will be equal to 22° 30', and will correspond with the true 25 6th part of the circle, or average space between the dots before laid down. The sector has consequently marked upon it eighteen intervals of 80' each, the first and last of which are subdivided into eight parts of 10' each. The fractional parts at each extremity are for the purpose of making the requisite coincidences with the former bisectional dots, and the intermediate sixteen divisions are the scale by which the true divisions are laid down.
Since this sector, though very correctly divided, may be liable to central error, its arc is made capable of a small adjustment, whereby 16§ths of its divisions are, by trial, made to correspond accurately with 1,thh part of the circle.
For cutting the divisions, the same apparatus is employed as was used by Ramsden in his dividing engine, but originally invented by Hindley, of York. These, together vn'th the two micrometer microscopes, constitute the whole apparatus to be employed.
The dividing point is first placed over that part of the circle at which the divisions are intended to be begun, while one of the microscopes is fixed accurately over the first of the 256 dots; and at the same time the first division of the sector is made to correspond with the wire of the second microscope.
The first division being now made, the circle is carried forward by a slow motion till the second division of the sector comes under the wire of its microscope, and the second division is now made upon
