Ill] OF THE CURVE OF ERROR 121 



thereto. If we pour a bushel of corn out of a sack, the outhne or 

 profile of the heap resembles such a curve ; and wellnigh every hill 

 and mountain in the world is analogous (even though remotely) to 

 that heap of corn *. Causes beyond our ken have cooperated to place 

 and allocate each grain or pebble; and we call the result a "random 

 distribution," and attribute it to fortuity, or chance. Galton 

 devised a very beautiful experiment, in which a slopmg tray is 

 beset with pins, and sand or millet-seed poured in at the top. 

 Every falhng grain has its course deflected again and again; the 

 final distribution is emphatically a random one, and the curve of 

 error builds itself up before our eyes. 



The curve as defined by Gauss, princeps mathematicorum — who 

 in turn was building on Laplace f — is at once empirical and 

 theoretical J ; and Lippmann is said to have remarked to Poincare : 

 *'Les experimentateurs s'imaginent que c'est un theoreme de 

 mathematique, et les mathematiciens d'etre un fait experimental ! " 

 It is theoretical in so far as its equation is based on certam hypo- 

 thetical considerations: viz. (1) that the arithmetic mean of a number 

 of variants is their best or likeliest average, an axiom which is 

 obviously true in simple cases — but not necessarily in all; (2) that 

 "fortuity" implies the absence of any predominant, decisive or 

 overwhelming cause, and connotes rather the coexistence and joint 

 effect of small, undefined but independent causes, many or few: 



* If we pour the corn out carefully through a small hole above, the heap becomes 

 a cone, with sides sloping at an "angle of repose"; and the cone of Fujiyama is an 

 exquisite illustration of the same thing. But in these two instances one predominant 

 cause outweighs all the rest, and the distribution is no longer a random one. 



t The Gaussian curve of error is really the "second curve of error" of Laplace. 

 Laplace's first curve of error (which has uses of its own) consists of two exponential 

 curves, joining in a sharp peak at the median value. Cf. W. J. Luyten, Proc. 

 Nat. Acad. Sci. xvm, pp. 360-365, 1932. 



I The Gaussian equation to the normal frequency distribution or "curve of 

 error" need not concern us further, but let us state it once for all: 



J, _ (xa-x)* 



^ V27T 



where Xf^ is the abscissa which gives the maximum ordinate, and where the maximum 

 ordinate, y^ = 1/^/(27t). Thus the log of the ordinate is a quadratic function of the 

 abscissa ; and a simple property, fundamental to the curve, is that for equally spaced 

 ordinates (starting anywhere) the square of any ordinate divided by the product of 

 its neighbours gives a scalar quantity which is constant all along (G.T.B.). 



