On dividing Instrmnents. 85 



finely divided scale of equal parts. The thing aimed at was, 

 to obtain a point upon the arc at the highest bisectional 

 nnniber oj'diulsions from 0, which in his eight feet quadrants 

 was 1024. = 83° eo'. The extent of the beam compasses, 

 with which he traced the arc upon the limb of the instrument 

 to be tlividcd, being set off npon that arc, gave the points 

 0" and 60"; which, being bisected, gave 30° more to com- 

 plete tlic total arc. A second order of bisections gave points 

 at 15° distance from each other; but that which denojid 75" 

 was most useful. Now, from the known length of' tlie ra- 

 dius, as m.casured upon the srale, the length of the chord 

 of 10' 'JO' was computed, taken off from the scale, and pro- 

 tracted from 75' forwards; and the chord of 4° 40', being 

 ascertained in the same manner, was set off from 90' back- 

 wards, meeting the cl^ord of 10' 20' in the continually bi- 

 sectional arc of 85' 20'. This point beina; foutid, the work 

 was carried on by bisections, and the chords, as they be- 

 came small enough, were set ofl" beyond rhis 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 nieans of dividing 

 bisectionally without one. His method I will briefly state as 

 follows, in the manner Vv'hich it would apply to dividing 

 a mural quadrant. The arcs of 60' and 3-0° give the total 

 arc as before; and let the last are of 30° be bisected, also 

 the last arc of 15°, and again the last arc of 7° 30'. The 

 two marks next 90° will now be 82' 30' and 86' 15', con- 

 sequently 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 S6° 15' also into three, the most 

 backward of the two marks will denote 85° 30'. Lastlv, if 

 we divide any one of these last spaces into five, and set ofF 

 one of these filth parts backwards from 85° 30', we shall 

 have the desired poinl :it 1024 divisions upon the arc from 0". 

 All the rest of the divisions which have been made in this 

 operation, which I have called marks because they should 

 be made as faint ao possible, must be erased ; for mv l)ro- 

 iher would not sufl'cr a njark to remain upon the arc to in- 

 terfere with his future bisections. 



Mr. vSineaton, in a paper to be more particularly noticed 

 prescntiv, justlv remarks the want of a iinitv of principle m 

 Mr. Bird's method; lor he proceeds partly on the ground of 

 the jirolrncted radiu**, and partly upon that of the computed 

 chord; which, a>s Smcalon observes, may or may not a'jree. 

 V 3 Bird, 



