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 knowledge 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 circular 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-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 trusting to probability.
We may often perceive that a series of measurements tends towards an extreme limit rather than towards a mean. In endeavouring to obtain a correct estimate of the apparent diameter of the brightest fixed stars, we 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, and J. Cassini, 5 or 6 seconds. Halley, Michell, and subsequently Sir W. Herschel came to the conclusion that the brightest stars in the heavens could not have real discs of a second, and were probably much less in diameter. It would of course be absurd to take the mean of quantities which differ more than 240 times; and as the tendency has always been to smaller estimates, there is a considerable presumption in favour of