Karl Pearson 
27 
This is a noruial surface wliich contains tlie prudiict terias. As we now interpret 
it we say tliat the .r's are correlated variates. And in this sense Gauss in 1823 
reached tlie normal surface of n correhated variates. But he does not seek to 
express all his relations in terms of the s.D.'s o-j., , o-c^, o-^^, ... and the correlations 
'■12, ^'ao •■• of these variates. These .r-variates are not for Ganss, nor for those who 
immediately followed him, the directly observed quantities. What he is seeking is the 
expression for a.^, or the probable error of an indirectly observed variate in terms of 
In this case A, B, C are ratios of minors and determinants of the a, /3, 7, ... which 
are Gauss' known quantities. His object therefore is to express ax not from 
direct observations but in terms of a, /B, y, ... through the sums of determinantal 
terms. 
Writers on Least Squares and Adjustment of Observations then take lu any 
function of Xi, x.^, ... x^, i.e. 
= F {X^, X„, ... Xn), 
express the relation in a linear form, i.e. 
to ~w = X, {a\ - + X,„ {xo - Xi) + 
and then, to find o-,/, go through lengthy analysis to determine 
Mean {x^ — Mean {x.2 — Ti:^)-, Mean (xi — J\) {x„ — Xo), etc. 
in terms of the original a, ^, y, There is not a word in their innumerable 
treatises that what is really being sought are the mutual correlations of a system of 
correlated variables. The mere using of the notation of the correlational calculus 
throws a flood of light into the mazes of the theory of errors of observation. There 
is much more in the theory of least squares than I have stated ; there are equations 
of conditions — the angle and side equations of geodesy, etc. — these only complicate 
the matter. The point is this : that the Gaussian treatment leads (i) to a non- 
correlated surfixce for the directly observed variates, (ii) to a correlation surface for 
the indirectly observed variates. This occurrence of product terms arises from the 
geometrical relations between the two classes of variates, and not from an organic 
relation between the indirectly observed variates appearing on our direct measure- 
ment of them. 
It will be seen that Gauss' treatment is almost the inverse of our modern 
conceptions of coiTolation. For him the observed variables are independent, for us 
the observed variables are associated or correlated. For him the non-observed 
variables are correlated owing to their known geometrical relations with observed 
variables ; for us the unobservable variables may be supposed to be uncorrelated 
causes, and to be connected by unknown functional relations with the correlated 
variables. In short there is no trace in Gauss' work of observed physical variables 
being — apart from equations of condition — associated organically which is the 
fundamental conception of correlation. 
