460 Averages. [CHAP. xvm. 



order to measure the errors. The errors in fact are not 

 known for certain but only probably calculated. This how 

 ever is not so much of a theoretic defect as it may seem at 

 first sight ; for inasmuch as we seldom have to employ these 

 methods, for purposes of calculation, that is, as distinguished 

 from mere illustration, except for the purpose of dis 

 covering what the ultimate average is, it would be a sort of 

 petitio principii to assume that we had already secured it. 

 But it is worth while considering whether it is desirable to 

 employ one and the same term for errors known to be 

 such, and whose amount can be assigned with certainty, and 

 for errors which are only probably such and whose amount 

 can be only probably assigned. In fact it has been proposed 1 

 to employ the two terms error and residual respectively 

 to distinguish between the magnitudes thus determined, that 

 is, between the (generally unknown) actual error and the ob 

 served error. 



22. (2) The other point involves the question to what 

 extent either of the first two tests (pp. 446, 7) of the close 

 ness with which the various results have grouped themselves 

 about their average is trustworthy or complete. The answer 

 is that they are necessarily incomplete. No single estimate 

 or magnitude can possibly give us an adequate account of a 

 number of various magnitudes. The point is a very im 

 portant one ; and is not, I think, sufficiently attended to, the 

 consequence being, as we shall see hereafter, that it is far 

 too summarily assumed that a method which yields the 

 result with the least error of mean square must necessarily 

 be the best result for all purposes. It is not however by any 

 means clear that a test which answers best for one purpose 

 must do so for all. 



It must be clearly understood that each of these tests is 



1 By Mr Merriman, in his work on Least Squares. 



