176 
ME. M. W. CEOFTON ON THE PEOOF OF 
generally received among mathematicians as expressing the law of frequency of single 
errors of observation, does not seem to have been distinctly given by any one of the three 
great philosophers Laplace, Gauss, and Poisson (who may be called the founders of 
the Theory of Errors) as being, in their opinion, the expression of that law. It has 
been erroneously supposed, as Leslie Ellis points out, that Gauss’s and Laplace’s 
proofs of the method of Least Squares depend upon that assumption. It is true that 
Gauss’s first method, in the ‘ Theoria Motus,’ does require it ; but he does not present 
that method as other than tentative and hypothetical : and later, in the ‘ Theoria Com- 
binationis Observationum,’ he says, speaking of the law of single errors, “ plerumque 
incognita est.” 
As, however, this law of error seems in our day to have been adopted by general 
consent, some inquiry into the grounds on which its validity rests will be appropriate 
here. And first I would remark that it can scarcely be maintained that any attempt 
hitherto made to establish this law independently of the hypothesis I have named in 
art. 1 has been successful. We may pass by Gauss’s proof in the ‘Theoria Motus,’ 
which shows that the law must hold if we take as an axiom that the arithmetical mean 
of several observations is the most probable result. Now this really is not an axiom, 
but only a convenient rule which is generally near the truth : this we see by considering 
any case in which we are certain that the errors do not follow the exponential law ; does 
the mind see here a priori that the rule does not give the most probable result \ It 
seems certain that we should have just the same confidence in it here as in any case; 
yet Gauss’s proof shows that it does not give the most probable result*. It should 
indeed be stated that Gauss himself (as might have been expected from that acute and 
accurate mind) is very far from asserting the above assumption to be an axiom ; conse- 
quently he does not give his proof as more than hypothetical. He only states that the 
rule is generally accepted — “ axiomatis loco haberi solet hypothesis.” A method of 
remarkable simplicity was given by Sir J. Herschel in a very interesting review of 
Quetelet’s ‘ Letters on Probabilityf ,’ which conducts to the same law of error by means 
of one or two bold assumptions ; but striking as the coincidence is, it can hardly be 
seriously viewed as a demonstration ; nor is it formally so presented by its distinguished 
author. However, the methods both of Gauss and Sir J. Herschel are of great interest 
to the natural philosopher, as showing that certain a priori mathematical assumptions 
of a very simple kind lead to the same law of error which reasoning based on a study of 
the facts which surround us also points out as expressing, at least approximately, what 
generally does occur in rerum naturd : though we can see no necessity that the facts 
* See Ellis, loc. cit. p. 207. 
t Edinburgh Eeview, July 1850. See a criticism by Leslie Ellis in the Philosophical Magazine, vol. xxxvii. 
Also Poole (Edinb. Trans, vol. xxi.) and Thomson and Tait (Natural Philosophy), who speak more favourably. 
M. Quetelet’s ‘ Lettres ’ will amply repay a perusal ; in connexion with our present inquiry, he points out that 
not only errors of observations, but the variations of many other fluctuating magnitudes, such as the stature of 
men, the temperature of the weather, &c. from their mean values, seem to follow the same law. If this be so, 
the inference seems legitimate that these divergences from the mean types, or errors of Nature herself, as they 
may bo called, are produced in each case, not by one or two, but by a vast number of hidden coexisting causes. 
