454 THE PRINCIPLES OF SCIENCE. 



The 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 entire absence of 

 any knowledge to the contrary. The selection of the 

 mean rests entirely upon the probability that wholly un 

 known 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 presumption 

 to the contrary, 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. 

 Thus we may certainly approximate to the length of the 

 circumference of a circle, by measuring the perimeters of 

 inscribed and circumscribed polygons of an equal and large 

 number of sides. The correct length of the circular line 

 undoubtedly lies between the lengths of the two perimeters, 

 but it does not follow that the mean is the best approxi 

 mation. It may in fact be shown upon mathematical 

 principles that the circumference of the circle is very 

 nearly equal to the perimeter of the inscribed polygon, 

 together with one-third part of the difference between 

 the inscribed and circumscribed polygons of the same 

 number of sides. Having this knowledge we ought of 

 course to act upon it, instead of upon vague grounds of 

 probability. 



We may often perceive that a series of measurements 

 tends towards an extreme limit rather than towards a 

 mean. Thus in endeavouring to obtain a correct estimate 

 of the apparent diameter of the brightest fixed stars, we 

 should find a continuous diminution in estimates as the 

 powers of observation increased. Kepler assigned to 

 Sirius an apparent diameter of 240 seconds ; Tycho Brahe 

 made it 126; Gassendi 10 seconds; Galileo, Hevelius, 



