438 THE PRINCIPLES OF SCIENCE. 



is infinitely more probable, diverge from it by an amount 

 which we must regard as error of unknown origin. Now, 

 to quote the words of Sir J. Herschel 6 , ' the probability of 

 that error is the unknown function of its square, i.e. of 

 the sum of the squares of its deviations in any two rect- 

 angular directions. Now, the probability of any deviation 

 depending solely on its magnitude, and not on its direc- 

 tion, it follows that the probability of each of these rect- 

 angular deviations must be the same function of its square. 

 And since the observed oblique deviation is equivalent to 

 the two rectangular ones, supposed concurrent, and which 

 are essentially independent of one another, and is, there- 

 fore, a compound event of which they are the simple in- 

 dependent constituents, therefore its probability will be 

 the product of their separate probabilities. Thus the 

 form of our unknown function comes to be determined 

 from this condition, viz., that the product of such functions 

 of two independent elements is equal to the same function 

 of their sum. But it is shown in every work on algebra 

 that this property is the peculiar characteristic of, and 

 belongs only to, the exponential or antilogarithmic function. 

 This, then, is the function of the square of the error, which 

 expresses the probability of committing that error. That 

 probability decreases, therefore, in geometrical progression, 

 as the square of the error increases in arithmetical.' 



Laplace's and Quetelet's Proof of the Law 

 of Error. 



However much presumption the modes of determining 

 the Law of Error, already described, may give in favour 

 of the law usually adopted, it is difficult to feel that the 



c ' Edinburgh Review,' July 1 850, vol. xcii. p. 1 7. Reprinted ' Essays,' 

 P- 399- This method of demonstration is discussed by Boole, ' Trans- 

 actions of Royal Society of Edinburgh/ vol. xxi. pp. 627-630. 



