Astronomical Methods of Observation. 131 



causing by that means a small error, which will he continually 

 repeated, and which, therefore, the principle of repetition can- 

 not cure. It is, I believe, owing to this cause th-at a con- 

 stant error of about 5", according to Baron Zach, may 

 remain in some instruments in a series of many hundred 

 observations made w^ith the repeating circle when the 

 clamping irons are imperfect. M. Biot goes on to say, that 

 the errors of the extreme readings at the commencement 

 and termination are much diminished, because the circle 

 has generally four verniers that are read separately, and of 

 which, the mean marks the commencement and termination 

 of the total with a great probability of accuracy. Finally, 

 the small error which still remains, notwithstanding these 

 precautions in the extreme readings, is distributed over the 

 entire arc measured on the limb, and therefore has an in- 

 sensible influence on the simple value of one observation, 

 when these observations are sufficiently multiplied. The 

 errors of the division, then, in the repeating circle itself are 

 also thus diminished by repetition, and the compensation 

 of errors is not the effect of probability, but of certainty. 



" To estimate the extent of this compensation, it may be 

 remarked, that our (French) repeating circles are generally 

 about 15 inches in diameter, and the error of division 

 cannot exceed 15". If the error would be reduced to half 

 a second after thirty observations, what would it become 

 after eighty or one hundred 1 What does it become after, 

 as has often been done, the series of different days are made 

 to succeed one another, without interruption, upon the 

 limb, so that the two errors of the extreme readings are 

 extended upon a total arc, which contains the simple arc 

 many thousand times ? The errors of division, then, in 

 this instrument become evanescent, and it is impossible 

 they can be entirely destroyed in the largest instruments, 

 if they are not repeaters. Never can the address of an 

 artist equal a mathematical proceeding." 



But there are other errors which are destroyed by the 

 principle of probabilities in the use of the repeating circle 

 that still remain in other instruments. Such are, the errors 

 of the level, which were small in the original repeating 

 circles, and in those later constructed still less, in which 

 the divisions of the level give immediately fractions of a 



K 2 



