354 M. Bessel's Examination of the Division 



portunity to express the admiration which this high perfec- 

 tion has forced from me. If the probable irregularity of a 

 diameter determined by six divisions resulting from this con- 

 tingent error, = + 0' , 133 *, which may be somewhat in- 

 creased by the errors of my own operations, be compared with 

 the above determined errors of division, there is, on the one 

 hand, no doubt that there is some regularity in them ; on 

 the other hand, the irregularities which occur in them are no 

 longer surprising. But if any advantage is to be derived from 

 their investigation for the reduction of the observations, the 

 irregular errors of division must be separated from the re- 

 gular ones ; for the latter only can be taken into account as 

 the circle is read off by verniers, which in almost every ob- 

 servation coincide with different divisions of the circle. 



In this respect there is an essential difference between the 

 circles with microscopes, and those with verniers : by ap- 

 plying my method one may entirely do away in the for- 

 mer the effects of the errors of division ; the latter are always 

 affected by the irregular part of the errors of division (as it 

 is not well practicable to examine every single division), and 

 allow only the attainment of a certain degree of precision, the 

 more accurate determination of which is of consequence for 

 the valuation of the final result. On the other hand, the ver- 

 niers have the advantage over the microscopes by their perfect 

 invariability, and their giving more accurate results when the 

 observations depend on divisions which have not been ex- 

 amined: with the microscopes the same contingent errors al- 

 waj's occur; with the verniers they change almost every day. 



I have endeavoured to separate the irregular errors of divi- 

 sion from the regular ones, and to determine the latter in such 

 a manner as to allow them to be taken into account. I have 

 best succeeded in this by considering the parts of the circle as 

 abscissas, and the errors as above found as ordinates ; and 

 by drawing freely with the hand, a curve agreeing with them 

 as nearly as was consistent with continuity, having before in- 

 creased the 48 errors of division above given, by means of 

 some more bisections, which, however, deserve less confidence. 

 I am far from believing that the ordinates of the curve will 

 correctly represent the law of the errors of division ; but I be- 

 lieve myself to be warranted in assuming that the application 

 of the curve will lead nearer to the truth, than if it were neg- 

 lected. Supposing the curve to be correct, the errors of 

 those divisions should be taken from it, which coincide with 

 the divisions of die vernier; but if on account of the possible 



