140 PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 



the error from each separate cause, provided positive and negative errors of equal 

 amount are equally likely ; but it is the complexity, not the sufficiency, of his 

 processes, which I think requires attention. 



4. Gauss' investigation is founded on the assumption, that the arithmetical 

 mean, of the results deduced from equally trustworthy observations, is the most 

 probable value of the quantity sought. So far as I can see, Ellis* has satisfactorily 

 shown that this, however apparently natural, is not justifiable as an a priori 

 assumption. In fact, it would seem that we have no right to assume that, be- 

 cause errors of equal magnitude and opposite signs are equally likely, their sum 

 will vanish in a large number of trials, any more than that the sum of their third 

 or fifth powers will vanish. Why the first powers should be chosen, appears to 

 arise from the extreme simplicity of the requisite operations ; yet, though com- 

 plexity of calculations is undesirable, it must be submitted to, if necessary for the 

 evolution of truth. The principle of the arithmetical mean has been adopted, 

 among a multitude of others equally likely, just as we might suppose a calculator 

 to insist on gravity varying as the direct distance instead of its inverse square, on 

 the ground that the problem of Three Bodies would then become as simple and 

 its solution as exact, as they are now complicated, and at best only approximate. 

 " La nature ne s^est pas embarrassee des diffiadtes d" 1 analyse, elle n'a evite que la 

 complication des moyens" in the words of Fresnel. 



5. It is with some hesitation that I communicate the present paper to the 

 Society ; for I have not devoted much time to the study of the Theory of Proba- 

 bilities ; and I know well how easy it is to fall into the gravest errors of reason- 

 ing on such a subject, from the fact that D'Alembert, Ivory, and many others, 

 have published investigations and proofs (sometimes in its most elementary 

 parts), which are now seen to be entirely fallacious. 



6. I proceed to show how I think the principle, above (§2) enunciated, may 

 be applied. The most direct method would be, of course, to assume any one set 

 of causes of error whatever, and to determine what will, in the long run, be the 

 chance of each separate amount of error as due to their joint action. Supposing 

 this to be determined, let us try to combine the probabilities of error from any 

 indefinite number of sets of possible causes ; and, if this process should lead to a 

 definite law of error, such will be the law to which, by an inverse application of 

 the Theory of Probabilities, we should expect each separate observation to be 

 subject. But this process, which is analogous to that of Laplace, though not iden- 

 tical with it, cannot easily be carried out, for it essentially involves in its first 

 steps the assumption of a law of error which it is the object of the investigation 

 to determine. We must try a less direct method. 



7. We shall, therefore, investigate what must be, in the long run, the chance 



* Cambridge Phil. Trans, viii. p. 205. 



