6 Mr. Ivory on the Method of the Least Squares, 



Thus the condition of the equilibrium of the levers makes 

 the sum of the squares of the errors a minimum ; and hence 

 we may infer that the errors themselves are contained within 

 the least limits on either side of the common fulcrum. If the 

 errors vary from the minimum, they cannot all decrease ; if 

 any of them decrease, others must increase; and some of 

 them at least must be greater than in the case of the minimum. 

 There appears, therefore, to be sufficient reason for preferring 

 the minimum of the squares of the errors as the most advan- 

 tageous solution of a system of equations of condition. 



Let us now place the matter in a different light. In any 

 system of errors of observation, supposing that there is no 

 constant cause of deviating from the truth, the total sum will 

 be equal to zero. At least, this will be the case if the obser- 

 vations be numerous, and if they embrace every possible va- 

 riety of circumstances. When a cause exists tending either 

 equally to augment, or equally to diminish, all the observa- 

 tions, the sum of the errors divided by their number will de- 

 termine the constant quantity affecting every observation. It 

 is evident that these considerations are independent of the 

 magnitude of the errors ; and therefore they can be of no use 

 in solving a system of equations of condition, where the ob- 

 ject is to make all the errors fall within the least possible 

 limits. Let us next consider the sum of the squares of a 

 system of errors, viz. 



e" + ^^ + e"^ + &c. 

 This sum is augmented by every observation, and it will 

 therefore increase indefinitely with their number. But as 

 every error lies between zero and a certain limit, if the sum 

 of their squares be divided by their number, the quotient, or 

 the mean of the squares, will also be contained between zero 

 and a limit; and it will approach more nearly to a determi- 

 nate value as the observations are more numerous. There- 

 fore, in a system of observations made in like circumstances, 

 the mean of the squares of the errors will be a quantity inde- 

 pendent of their number, varying in its magnitude only as 

 the errors are more or less considerable, and affording with 

 some accuracy a measure of the precision of the observations. 

 In several sets of observations made for the same purpose by 

 different observers and in different circumstances, that one in 

 which the mean of the squares of the errors is least must be 

 considered as possessed of the greatest degree of jirecision, 

 and would deserve the preference. Now, in solving a system 

 of" ccjuations of condition in several different ways, the errors 

 will accjuire different magnitudes, just as happens in several 

 sets of observations of unequal degrees of precision. We 



will, 



