40 Mr. F. Galton on Statistics by Intercomparison, 



instances, where measurement was possible, that the latter con- 

 form very fairly, within the limits of ordinary statistical inquiry, 

 to calculations based on the (exponential) law of frequency of 

 error. It is a curious fact, which I shall endeavour to explain, 

 that in this case a false hypothesis, which is undoubtedly a very 

 convenient one to work upon, yields true results. 



In illustration of what occurs in nature, let us consider the 

 causes which determine the size of fruit. Some are important, 

 the chief of which is the Aspect, whose range of influence is too 

 wide to permit us to consider it in one of the simple alternatives 

 " good" or " bad." It is no satisfactory argument to say that 

 variations in aspect are partly due to a multitude of petty causes, 

 such as the interposition of leaves and boughs, because, so far 

 as they depend on well-known functions of altitude and azimuth, 

 they cannot be reduced to a multitude of elementary causes. 

 There has been much confusion of ideas on this subject, and 

 also a forgetfulness of another fact — namely, that when we 

 once arrive at a simple alternative, there our subdivision of 

 causes must stop, and we must accept that alternative, how- 

 ever great may be its influence, as one of the primary ele- 

 ments in our calculation. 



In addition to important elements, there are others of small, 

 but still of a recognizable value, such as exposure to prevalent 

 winds, the pedigree of the tree, the particular quality of the soil 

 on which it stands, the accident of drains running near to its root> 

 &c. Again, there are a multitude of smaller influences, to the 

 second, third, and fourth orders of minuteness. 



I shall proceed to define what I mean by " small ; " then I 

 shall show how this medley of causes may admit of being theo- 

 retically sorted into a moderate number of small influences of 

 equal value, giving a first approximation to the truth ; then how, 

 by a second approximation, the grades of the binomial expansion 

 thence derived become smoothed into a flowing curve. Lastly, 

 I shall show by quite a different line of argument that the expo- 

 nential view contains inherent contradictions when nature is ap- 

 pealed to, that the binomial of a moderate power is the truer one, 

 and that we have means of ascertaining a limit which the number 

 of its elements cannot exceed. My conclusion, so far as this 

 source of difficulty is concerned, is that the exponential law ap- 

 plies because it nearly resembles the curve based on a binomial of 

 moderate power, within the limits between which comparisons are 

 usually made. 



We observe in fig. 2 how closely the outline of an exponential 

 ogive resembles that of a binomial of a very moderate number 

 of elements, within the narrow limits chiefly used by statisticians. 

 The figure expresses a series of 1000 objects marshalled accord- 



