30 Arrangement and Formation of the Series. [CHAP. n. 



central values, for those that is which are situated most 

 nearly about the mean. With regard to the extreme values 

 there is, on the other hand, some difficulty. For instance 

 in the arrangements of the heights of a number of men, 

 these extremes are rather a stumbling-block ; indeed it has 

 been proposed to reject them from both ends of the scale 

 on the plea that they are monstrosities, the fact being that 

 their relative numbers do not seem to be by any means 

 those which theory would assign 1 . Such a plan of rejection 

 is however quite unauthorized, for these dwarfs and giants 

 are born into the world like their more normally sized 

 brethren, and have precisely as much right as any others 

 to be included in the formulae we draw up. 



Besides the instance of the heights of men, other classes 



not be supposed that all specimens of 

 the curve are similar to one another. 

 The dotted lines are equally specimens 

 of it. In fact, by varying the essen 

 tially arbitrary units in which x and 

 y are respectively estimated, we may 

 make the portion towards the vertex 

 of the curve as obtuse or as acute as 

 we please. This consideration is of 

 importance; for it reminds us that, 

 by varying one of these arbitrary 

 units, we could get an exponential 

 curve which should tolerably closely 

 resemble any symmetrical curve of 

 error, provided that this latter re 

 cognized and was founded upon the 

 assumption that extreme divergences 

 were excessively rare. Hence it 

 would be difficult, by mere observa 

 tion, to prove that the law of error 

 in any given case was not exponen 

 tial; unless the statistics were very 

 extensive, or the actual results de 



parted considerably from the expo 

 nential form. (2) It is quite impos 

 sible by any graphic representation 

 to give an adequate idea of the exces 

 sive rapidity with which the curve 

 after a time approaches the axis of x. 

 At the point jR, on our scale, the curve 

 would approach within the fifteen- 

 thousandth part of an inch from the 

 axis of x, a distance which only a very 

 good microscope could detect. Where 

 as in the hyperbola, e.g. the rate of 

 approach of the curve to its asymp 

 tote is continually decreasing, it is 

 here just the reverse ; this rate is 

 continually increasing. Hence the 

 two, viz. the curve and the axis of x, 

 appear to the eye, after a very short 

 time, to merge into one another. 



1 As by Quetelet: noted, amongst 

 others, by Herschel, Essays, page 

 409. 



