889 On a Method of examining 



...e^'"'"^"" are all negative^ then it will appear that ^^~^x 



2p<? will be the greatest error that can be committecfin de- 

 termining the value of any arc by adding together the values 

 of the (p) smaller arcs of which it is compounded. For 

 instance, if the interval betwixt the micrometer and- the 

 microscope comprehends an arc of 60°, as marked upon 

 the instrument, and this arc is measured against every suc- 

 ceeding arc of 60° in the whole circle, we shall have the 

 greatest error that can be conynitted in deducing the arc 

 of 120° from the addition of the two first arcs of 60% 



equal to — ^ x 2 x 2e = 2*66 e. After these remarks, 



we may proceed to consider how the remaining divisions 

 upon the circle may be examined with the lta>t probable 

 error, and to ascertain the amount of the greatest to which 

 the process can in any case be liable. 



Let the arc of 30° be now measured against every suc- 

 ceeding arc of 30° in the first, third, fourth, and sixth arcs 

 of 60°, and let the length of each be determined from a 

 separate comparison with the arc of 60^, in which it is 

 comprehended, and not from a general comparison with all 

 the four. The arc of 15° must then be measured against 

 every succeeding arc of 15° in the first, third, fourth, sixth, 

 seventh, ninth, tenth, and twelfth arcs of 30% and the 

 value of each deduced from a comparison with the arc of 

 30^, in which it is contained. When this is done, we 

 shall have determined the length of every succeeding arc of 

 15% of the first arcs of 30, 45, 60, 75 (= 60 + 15), 90, 

 105 (= 90 + 15), 120 (= 60 + 60), 135 ( = 90 + 45), 150 

 (= 120 + 30), 165 (= 150 + 15), and 180^ in each semi- 

 circle. 



We must next measure the arc of 5^ against every suc- 

 ceeding arc of 5® in the whole circle, and deduce the values 

 of the first, and of the sum of the first and second, in each 

 succeeding arc of 15% from a comparison with the arc of 

 15" in which they are contained. We must then proceed 

 to determine the values of the first arc of 3° in each 15% 

 and of its multiples the arcs of 6, 9, and 12°. We must 

 also put down the value of the last arc of 3° in each arc 

 of 15°, and then deduce the values of the first and last 

 arcs of 1° in each ?rc of 15°, tVom a comparison with the 

 arc of 3° in which they are respectively cwnlained. 



We shall now have measured in each arc of 15° the first 

 arcs of 1,3, 5, 6, 9, 10, 12°, and by taking the la&t arc of 

 one degree, which has likewise been determined, from the 



arc 



