ON THE METHOD OF LEAST SQUARES. 35 



The second demonstration may be thus briefly stated. 



The values of different sets of observations might be com- 

 pared if we knew the average error in each set, or if we knew 

 the average value of the squares of the errors in each. In either 

 case that would be the best set of observations in which the 

 quantity taken as the measure of precision was the smallest. 



Similarly, by assigning different values to the unknown 

 quantities x, y, &c. involved in a system of equations of condi- 

 tion, we can make it appear that the mean of the squares of the 

 errors has a greater or less value. Therefore as of sets of obser- 

 vations, that is the best in which this quantity is least ; so of 

 different sets of results deduced from one set of observations, the 

 same is also true ; and therefore the sum of the squares of the 

 apparent errors is to be made a minimum. 



There seems to be involved in this reasoning a confusion of 

 two distinct ideas; the precision of a set of observations is 

 undoubtedly measured by the average of the errors actually 

 committed, and if we knew this average, we should be able 

 to compare the values of different sets of observations. But 

 it is not measured by the average of the calculated errors, 

 namely, those which are determined from the equations of 

 condition when particular values have been assigned to x, y, &c. 



The problem to be solved may be stated thus. Given that 

 the single observations of which the set is composed are liable 

 to a certain average of error, to combine them so that the re- 

 sulting values of the unknown quantities may be liable to the 

 smallest average of error. 



This problem Laplace and Gauss have both solved. Their 

 solutions differ, because they estimated the average error in 

 different manners. 



But how are we justified in assuming that to be the best 

 mode of combining the observations which merely gives the 

 appearance of precision not to the final results, but only to the 

 individual observations, and which, with reference to them, gives 

 no estimation of the probability that this appearance of accuracy 

 is not altogether illusory ? 



The third of Mr Ivory's demonstrations is not, I think, more 

 satisfactory than the other two. 



The kind of observations to which the method of least 



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