DISCOVERY BY MEANS OF CALCULATIONS. 611 



applicability of it depends upon the readiness with which 

 our eye detects regularity or irregularity of forms. It is 

 used to detect inexact results, to correct them, and also to 

 discover the laws of change which true results obey. By 

 the method of means we are enabled to get rid of various 

 errors and inequalities, by taking the arithmetical mean of 

 a large number of observed quantities, because in a large 

 number of observations ordinary irregularities compensate 

 each other ; it also thus enables us to discover very much 

 smaller errors and influences than those which balance 

 each other. In the method of least squares we take that 

 mean which, according to the doctrine of chances, is the 

 most probable one, viz., that which is taken according to 

 the condition, that the sum of the squares of the errors of 

 observation shall be the least possible which the law of the 

 facts permits. 1 In the method of residues w,e subtract 

 from the quantities supplied by observation the quantity 

 necessary to satisfy previously-known laws operating in the 

 case, and then examine the residue to discover the chief 

 law which it obeys. We next subtract from the residue 

 the quantity necessary to satisfy this last law, and thus 

 obtain a second residue, to be treated in a similar manner ; 

 and thus bring each law into view in succession. The 

 value of the method depends chiefly upon the fact that the 

 mean effects of the successive laws of variation are less and 

 less in magnitude, so that the law in question is not pre- 

 vented being prominent by the undiscovered laws. Both 

 in the method of curves and residues, after the arguments 

 of the chief irregularities are known, it is comparatively 

 easy to find the remaining ones. 



'The law of continuity consists in this proposition : 



1 Both in the method of means and of least squares we require to 

 know previously the arguments of the irregularities of which we are 

 in search. 



B B 3 



