26 
Notes on the History of Correlation 
We observe or measure directhj a certain number of quantities a, h, c, d, — 
Each of these quantities is supposed by Gauss to be independent and to follow the 
normal law. The combined probability of the system* is accordingly 
or the product of the independent probabilities, where (t„ , a/,, a,., ... are the variability 
in errors of a, b,c, ... and a, h,c., ... the means. This probability will be a maximum 
when 
a, b, , 
is a minimum. This is really the principle of weighted least squares. Its validity 
depends upon the normal law of distribution of error. Without this law holding it 
may be a utile method, but we have no means of proving it the ' best.' 
The investigator in Gauss' case is, however, not interested in the quantities 
observed, but in certain indirectly ascertained quantities a\, x.^y.-Xn which are 
functions of them. Thus 
a\ = f\ (a, h, c, ...), 
a;., = fi ((I, h, c, ...), 
where /d /oi ••• are known functions. Now Gauss cannot as a rule express from 
these general equations a, h, c, ... in terms of x^, ,t.2, . . . x^. 
He assumes that all of them differ slightly from their mean or ' true ' values and 
accordingly expands by Taylor's theorem and reaches the resultf 
- rf^i = a, (a - a) + /3i (6 - i) + 7i (c - r) + 
X. - X.2 = tta (rt - (7) + /?., (6 - 6) + 7., (c - c) + . . . , 
where the a, B, 7, ... are , , ... and can be ascertained a priori. Clearly 
' da db do ^ •' 
Gauss supposes that a linear relationship is adequate, in other woi'ds he replaces 
statistical differentials by mathematical differentials, a step he does not really justify. 
From these linear equations we can find the a — a , h — h, c — c , . . . m terms of 
the indirectly observed variables x^ — x^, x^ — x^, x-j — x-i, ... by solution in deter- 
minantal form, say 
a~a = {x, + -^i (a'. - X2) + G, (x-, - x.) 
h-l = A.2 (xj - a'l ) + Bo {x., - -f C„ {x-i - x-i) .... 
Substituting in u- we find 
n\,, x„ .„ = S (^j {X, -x,f + S (^^) {X, - x^f + 28 i^y^h - ^\) {x^-x;)^.... 
Hence the probability of x^, x.,, ... occurring is 
* I use throughout notation which I assume now-a-days to be more familiar than that of Gauss, 
t t7, h, c, ... are actually in Gauss' method approximate or guessed solutions not means, but this does 
not affect the seneral nature of the discussion. 
