34 On Equations of Condition for a Quadrilateral. [No. 121. 



to very lengthy calculations. But though the common rule for finding 

 the probability of error on a number of observations be as good as, or 

 perhaps better than, any other at present known, I think it may be 

 shewn that after all it has only a chance of being right, and is far from 

 certainty in all cases. It may be a pretty good approximation when 

 the number of observations is great, but when the number is small it 

 seems somewhat dubious ; at least, when the number is a minimum it is 

 palpably false, and it is not likely that it should become false per saltum. 



The rule commonly given, is to take the arithmetical mean of all the 

 observations, and the difference between this and each observation ; 

 then squaring each of these differences, take the square root of their 

 sum ; which root divided by the number of observations gives the pro- 

 bability of error ; the reciprocal of which gives the proportional weight 

 due to each. 



I have long sought, but never met with, a demonstration of this rule. 

 According to it, however great be the number of observations, if they 

 differ among themselves by any quantity, however small, there will be 

 a probability, however small, of error; and therefore the result must 

 fall short of certainty. But if there be any number of observations 

 agreeing among themselves, or even only one observation, there is no 

 probability of error : so at least says the rule, whereas common sense 

 says in the latter case the probability of error is very great, though 

 we have no means of making a better of it. 



Hence I hold that the rule commonly given for finding the probability 

 of error on a set of observations, though in general a pretty good prac- 

 tical rule, is not a mathematical truth : and I would not, on the faith of 

 its being such, build a cumbrous computation to obtain a result not 

 much, if at all better than that which may be obtained with one-tenth 

 part of the labor : for such I believe would be the disproportion be- 

 tween the combination of these equations according to the method above 

 indicated, and that by their weights as directed by writers on probabili- 

 ties. 



As to the practical difference in the results by these two methods of 

 adjustment, I cannot speak from actual trial; but I believe it would 

 rarely exceed one or two hundredths of a second ; and if we recollect 

 that it is amazingly difficult with ordinary, or even with extraordinary 

 instruments, to observe to within ten times the greatest of these quan- 



