378 THE PRINCIPLES OF SCIENCE. (CHAP. 



equivalent to the two rectangular ones, supposed concur- 

 rent, and which are essentially independent of one another, 

 and is, therefore, a compound event of which they are the 

 simple independent constituents, therefore its probability 

 will be the product of their separate probabilities. Thus 

 the form of our unknown function cornes 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 antilogarithniic 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. 



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 

 arguments are satisfactory. The law adopted is chosen 

 rather on the grounds of convenience and plausibility, than 

 because it can be seen to be the necessary law. We can 

 however approach the subject from an entirely different 

 point of view, and yet get to the same result. 



Let us assume that a particular observation is subject 

 to four chances of error, each of which will increase the 

 result one inch if it occurs. Each of these errors is to be 

 regarded as an event independent of the rest and we can 

 therefore assign, by the theory of probability, the compara- 

 tive probability and frequency of ea^h conjunction of errors. 

 From the Arithmetical Triangle (pp. 182-188) we learn that 

 no error at all can happen only in one way ; an error of 

 one inch can happen in 4 ways ; and the ways of happening 

 of errors of 2, 3 and 4 inches respectively, will be 6, 4 and 

 I in number. 



We may infer that the error of two inches is the most 

 likely to occur, and will occur in the long run in six cases 

 out of sixteen. Errors of one and three inches will be 

 equally likely, but will occur less frequently ; while no 

 error at all or one of four inches will be a comparatively 



