xvii.] THE LAW OF ERROR 389 



dicular line in falling down a deep pit, have been cited by 

 Encke l as an interesting illustration of the Law of Error. 

 The mean deviation was 5-086 lines, and its probable error 

 was calculated by Encke to be not more than -950 line, 

 that is, the odds were even that the true result lay between 

 4-136 and 6-036. As the deviation, according to astrono- 

 mical theory, should be 4 - 6 lines, which lies well within 

 the limits, we may consider that the experiments are 

 consistent with the Copernican system of the universe. 



It will of course be understood that the probable error 

 has regard only to those causes of errors which in the long 

 run act as much in one direction as another ; it takes no 

 account of constant errors. The true result accordingly 

 will often fall far beyond the limits of probable error, owing 

 to some considerable constant error or errors, of the ex- 

 istence of which we are unaware. 



Rejection of the Mean Result. 



We ought always to bear in mind that the mean of any 

 series of observations is the best, that is, the most probable 

 approximation to the truth, only in the absence' of know- 

 ledge to the contrary. The selection of the mean rests 

 entirely upon the probability that unknown causes of error 

 will in the long run fall as often in one direction as the 

 opposite, so that in drawing the mean they will balance 

 each other. If we have any reason to suppose that there 

 exists a tendency to error in one direction rather than the 

 other, then to choose the mean would be to ignore that 

 tendency. We may certainly approximate to the length 

 of the circumference of a circle, by taking the mean of the 

 perimeters of inscribed and circumscribed polygons of an 

 equal and large number of sides. The length of the cir- 

 cular line undoubtedly lies between the lengths of the two 

 perimeters, but it does not follow that the mean is the 

 best approximation. It may in fact be shown that the 

 circumference of the circle is very nearly equal to the 

 perimeter of the inscribed polygon, together with one -thkd 

 part of the difference between the inscribed and circum- 

 scribed polygons of the same number of sides. Having 



1 Taylor's Scientific Memoirs, vol. ii. pp. 330, 347, &c. 



