M. Bessel on the Divisions of Reichenbach's Circle. 351 



From these observations the most probable values of the 

 errors of division are to be deduced; their number is 24, but 

 \J/w being equal to vJ/(?<+180) they are reduced to 12, one of 

 which may be assumed arbitrarily; I supposed 4<0 = v{/ 180 = 0, 

 and obtained, by a solution adapted to the present case of the 

 1 1 equations resulting from the method of least squares, the 

 following values 



In order to appreciate the accuracy of these determinations, 

 I remark that every error of division is as accurately deter- 

 mined as if it had been derived from 30-42 observations of an 

 angle between two diameters (referred to six divisions) or 

 91-26 single positions of the microscopes; I find by all obser- 

 vations the probable uncertainty of a single reading of the 

 microscopes to be = + 0"-1825; and therefore the probable 

 error of ty(n 15°) = + 0"-0191.* This great accuracy is a con- 

 sequence of the clearness and neatness of the divisions, the 

 excellence of the microscopes, the regularity of their screws, 

 and the solidity of their fastening ; but at the same time the 

 frequent repetition of the determination of the angles and their 

 numerous crossings was necessary, in order to carry the ac- 

 curacy so far beyond the limits of the accuracy of a single 

 observation. All errors in these observations being contin- 

 gent, the accuracy may be carried to an unlimited extent, 

 certainly much further than the artist has carried it in making 

 the divisions : the astronomer who examines his instrument 

 according to my method, has the great advantage over the 

 artist who divided it, that he can repeat the operation as often 

 as he pleases ; whereas the artist has to depend on the single 

 operation of setting the apparatus for cutting the divisions, which 

 may be done according to the quality of that apparatus with 

 greater or less accuracy, but never with absolute correctness. 



In order to determine the angles of the form n 15° + 7° 30', 

 I have placed the microscopes at angles of 7° 30' to each other, 

 and have derived the errors from a comparison with the pre- 

 ceding and the following numbers of the foregoing table : in 



( — , i-O' 0191. — IiiAN%r.. I 



this 



