54 REMARKS ON AN ALLEGED PROOF OF 



Motus Elliptic^ Gauss undoubtedly does assume that the arith- 

 metical mean is the most probable value in the case of direct 

 observations of a single element. From this assumption, he 

 shows that the probability, that the magnitude of an error lies 

 between x and x + dx, must be 







N 



TT 



h being an indeterminate constant. It follows from this, that 

 the results of the method of least squares are always the most 

 probable values that can be assigned to the unknown elements. 

 Without referring to the Theoria Motus, you can see the de- 

 tails of Gauss's reasoning in a paper by Bessel, of which a 

 translation appeared in Taylor's Scientific Memoirs. The re- 

 viewer is right in saying that Gauss was not entitled to assume 

 that the arithmetical mean is the most probable value. But 

 when he speaks of this as a thing to be proved, and not as- 

 sumed, we are led to suppose that he believes that subsequent 

 writers have actually proved it. In truth this appears, not 

 only from his statements, but also from the illustrations of 

 which he has made use. Thus he states that if shots are fired 

 at a wafer which is afterwards removed, and we are asked to 

 determine from the position of the shot-marks the most pro- 

 bable position of the wafer, " the theory of probabilities affords 

 a ready and precise rule, applicable not only to this but to far 

 more intricate cases;" and he goes on to say that it may be 

 shown that the most probable position of the wafer is the 

 centre of gravity of the marks. Now this result is only then 

 true when the law of probability of error, which is implied in 

 Gauss's assumption, really obtains ; so that, according to the 

 reviewer, the demonstration of the principle of least squares 

 must amount to showing that this law obtains universally ; or, 

 which is the same thing, that the arithmetical mean is always 

 the most probable value in the case of direct observations of 

 a single element. If this can be proved, it is doubtless a very 

 curious conclusion; but it is at any rate certain that Laplace 

 has not proved it, of whom however the reviewer asserts that 

 he has given a rigorous demonstration of the principle of 

 least squares. From one end of Laplace's great work to 

 the other, there is nothing to justify the assertion that the 



