of Error and Correlated Averages, 525 



more than what is deducible from Taylor's theorem, because 

 for short distances from the maximum of F(#, y) (the surface 

 of frequency for x and y) the value of y most likely to 

 attend any assigned value of x may, in general, be regarded 

 as a linear function of x. One might as well attribute to 

 Taylor's theorem the fulfilment of the law of error in the 

 case of a single variable, on the ground that any continuous 

 curve symmetrical about a certain point may be identified for 

 short distances from that point with a probability-curve. 

 That consideration alone would not have enabled us to predict 

 the fulfilment of the law up to a sensible distance from the 

 centre — that exact consilience between the quartiles, octiles, 

 deciles, &c. of theory and of fact which has been observed by 

 Mr. Galton and others. Those verifications allow us to pre- 

 sume the existence in rerum natural of elementary agencies in 

 such numbers and of such magnitude as to generate a proba- 

 bility-curve of sensible dimensions. From that presumption 

 we may reason down to the genesis of a probability-surface 

 also of sensible dimensions. And this conclusion is confirmed 

 by specific experience : the sort of correlation which is dedu- 

 cible from that hypothesis having been, in fact, observed by 

 Mr. Galton and Prof. Weldon. There is required here not 

 only the use of Taylor's theorem, but also the presence of the 

 essential condition of the law of error; the action of numerous 

 independently variable elements. 



In fine, the theorem of this section completes the answer 

 which was given in a former section to the objection brought 

 by Cournot and Prof. Westergaard against Quetelet's prin- 

 ciple of the Mean Man; in substance, that the mean organs 

 obtained by averaging a number of specimens might not fit 

 each other. It is now seen that the objection in the form in 

 which it has been stated * implies a very rudimentary con- 

 ception of " correlation," viz. that to assigned values of some 

 organs the corresponding value of another organ is in each 

 particular instance and not merely on an average a function of 

 those assigned values. This sort of exact correlation probably 

 exists in natural history only with regard to those geometrical 

 relations between lines and angles which the objectors indi- 

 cate f« The objection in this form has been answered in the 



* Ante, p. 437. 



t Their instance, it will be remembered, was : two sides of a right- 

 angled triangle and the base. As right angles are not very prevalent in 

 organic nature, a happier instance might be : for three " organs," two 

 sides a and b and the contained angle C ; and, for the correlated organ, 

 the base c j which = V«'" + 6 2 — 2a6cosC. A concrete example of such 

 correlation is the triangle formed by the breasts and the throat (Qaatelet, 

 Anthropometric, p. 224:). 



Phil. Mag. S. 5. Vol. 34. No. 211. Dec, 1892. 2 



