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however often we may repeat the observation, the mean will still 

 remain 40''. Yet all the while the true angle may be 41, 42, 43, 

 or even 44" ; and it would appear that with such an instrument, an 

 infinity of observations could give us no better a result than a single 

 one. 



But if I apply to the same purpose a ten-inch circle carrying a 

 twelve -inch telescope with which 10" can only be estimated, and 

 having its graduation pushed somewhat beyond the limits of work- 

 manship, I find my measurements to fluctuate from 20" to 60". 

 Then the advantage of multiplied observations becomes apparent, 

 and twenty operations give us a result different from that given by 

 ten ; and if we admit this system of averaging, we are carried to 

 the absurd conclusion that a small instrument gives more exact 

 results than a large instrument does. 



In truth, this averaging of multiplied observations is a fallacy ; 

 if the results agree, the averaging is useless ; if they do not agree, 

 their discordance affords evidence that the means employed are in- 

 sufficient to procure the accuracy aimed at. 



The same remarks apply to time-observations. To observe the 

 meridian passage of a star we note the instant of its appulse to 

 each of the vertical wires, add the results and take the mean, so 

 that if there be a considerable number of wires, great precision is 

 expected. Now, at each wire we can only, and with hesitation, 

 note the time to the nearest tenth of a second. Suppose that we 

 can do so absolutely, and imagine the wires so placed, that the time 

 of passing from the one to the other is exactly a number of tenths. 

 Then, if the true time of appulse to the first wire w^ere, -04", the 

 observed would differ from the true by .04" at each one of the wires, 

 and the mean would err also by that quantity. Now, there are 

 many declinations which give the interval of passage from wire to 

 wire an exact number of tenths, so that, even supposing the ear 

 perfect to the nearest tenth of a second, there must be many cases 

 in which the average may be -04" wrong. 



Taking into account the various sources of error in the gradua- 

 tion and adjustment of instruments, we can scarcely assume that the 

 declination of any star is known certainly to half a second of arc, or 

 its right ascension to the twentieth second of time ; and it appears 

 that the true use of multiplied observation is to guard against blun- 

 ders in reading off, and to indicate the degree of confidence which is 



