the Theory of Probabilities. 59 



have any pretensions to these qualifications agree in thfe same 

 result is certainly a very remarkable circumstance, and one which 

 can hardly fail to excite some degree of expectation that this 

 result will turn out to be the true one, if the problem should 

 ever be really solved. I shall not go through an examination of 

 instances in illustration of the above remark ; but there are two 

 which I must mention briefly. 



The first is the Edinburgh Reviewer^s proof, commented upon 

 by Mr. Ellis. Of course it was easy to annihilate it, considered 

 as a professed demonstration. But if it had only pretended to 

 be what it really is, a proof founded upon an assumption (of the 

 independence of errors in directions at right angles to one an- 

 other) which is simple and not more arbitrary than the assump- 

 tions made in other proofs, while it leads to the result with 

 remarkable ease and directness, it would, I think, have deserved 

 to be treated with respect. It is to be regretted that the Re- 

 viewer should have failed to see, or at least to point out, its real 

 character. 



The second instance is a proof proposed by myself some years 

 ago in an Essay published by the Ashmolean Society, and since 

 abridged in Liouville^s Journal, vol. xv. This proof depended 

 upon a more complete and systematic development of the analogy 

 between the balance of evidence and the balance of forces, than 

 had been before attempted, and was published chiefly on account 

 of the interest which belongs (at least in my estimation) to all 

 such analogies. I was therefore not concerned to point out, and 

 indeed did not till lately clearly apprehend, what was the assump- 

 tion really involved in it. This assumption is, that the knowledge 

 gained from a number of observations is the same in kind as that 

 gained from a single observation. It is easy to make this the 

 foundation of the theory, treated according to the ordinary 

 method; to begin, namely, by assuming that the function ex- 

 pressing the law of facility of error of the mean of two observa- 

 tions, is of the same form as that which expresses the law for the 

 individual observations. I am inclined to think this assumption 

 in itself more simple and natural than any other ; but this is a 

 matter of opinion. 



I may add, that in the first paragraph of the preface to the 

 English edition of the Essay just mentioned, I committed the 

 fallacy which I have endeavoured to explain in the former part 

 of this paper, of confounding the case in which the actual law 

 of facility is unknown, with the case in which it is known. 



Oxford, May 23, 1851. 



P.S. Since Part II. of these remarks was printed, I have, 

 through the kindness of Professor De Morgan, received his 



