of using the ThcodolUe, 181 



port of the method under review ; but it is quite obvious that 

 the circumstance of one term being measured with extra pre- 

 cision, does not entitle it to be counted several times in striking 

 the mean. 



From the preceding investigation, we can easily discover 

 what is the proper arrangement of readers. In the first place, 

 all the readings must be regarded as equally entitled to atten- 

 tion, so that their simple average must be taken. In the second 

 place, the reversion of the telescope must not cause any. two of 

 them to exchange places ; and thirdly, the error of excentricity 

 must be corrected. Now the error arising from the excentricity 

 is a function of the sum of perpendiculars let fall upon lines 

 touching the limb at the readers ; while there is this property of 

 equilateral polygons, that the sum of perpendiculars let fall upon 

 their sides is the same from whatever point they may proceed ; 

 so that, if the readers be uniformly arranged around the limb, 

 the error arising from excentricity must be constant, and there- 

 fore must be eliminated from each observation. On the whole, 

 then, it is best to have an odd number of readers disposed uni- 

 formly around the circle. 



Mere opinion has too long held the place of accurate study 

 in the construction of angular instruments. In particular, the 

 question whether the method of repetition, or that of frequent 

 readings, be preferable, has been discussed with almost national 

 warmth. Repeated observations are French, single observa- 

 tions are English, as if there be national scientific creeds. Let 

 us inquire, by help of the same strict analysis, which of the two 

 gives the greatest probability of precision. 



The errors of excentncity and of collimation need not be 

 counted, since these are guarded against in a proper observa- 

 tion according to either metliod. The errors of graduation 

 and of fixing alone remain to be considered. Let the expecta- 

 tion of error on one observation with all the readers, and with 

 reversion, be E, that on the last repeated arc is just E ; so that 



if A; be the number of repetitions, | is the expectation of error 



on the result ; whereas if k single observations had been made, 



the expectation of error would have been -E^. An example 



will shew the matter in a clear light. Suppose an instrument 



