to the 3Iethod of Least Squares. 166 



lu his first proof Adrain had found for the probability of the error 



mx^ 



C-\- 



X in the observed value a the expression e 2a ^ and had shown 



that the most probable values of the observed quantities a, b, c, . . . . 



whose errors are a-, y,z, . . . must satisfy the condition 



^2 y2 ^2 



— + ^H h . . . . =: a minimum. 



a be 



This principle he applies to four problems ; the first showing that the 

 arithmetical mean is a particular case of the method, the second to 

 determine the most probable position of an observed point in space, 

 which is shown to be " precisely in the centre of gravity of all the 

 given points ;" the third " to correct the dead reckoning at sea," and 

 the fourth " to correct a survey." 



As remarked by Abbe " we must credit Dr. Adrain with the inde- 

 pendent invention and application of the most valuable arithmetical 

 process that has been invoked to aid the progress of the exact sci- 

 ences." — Atner. Jour. ScL, 1871, Vol. I, p. 415. 



1809 Gauss. ' Determiuatio orbitae observationibus quotcunque 

 maxime satisfacientis.' T/ieoria motus corpornm coelestium .... 

 (Hamburgi, 4to), Lib. II, Sect. Ill, pp. 205-224. —French transla- 

 tion, see 1855 Bektraxd. — Englisli translation by Davis (Boston, 

 1858, 4to), pp. 249-273. — German trans, by Haase (Hannover, 1865). 



That demonstration of the Method of Least Squares usually called 

 Gauss's proof or Gauss's first proof is here presented. Assuming 

 that the arithmetical mean of direct observations is the most proba- 

 ble value of the measured quantity, it deduces that the law of facil- 

 ity of error is given by 



cp{x) =z ce~^^^^ 



from which the principle of Least Squares at once follows. This 

 proof has been adopted by the majority of l)ooks on the subject • see 

 for instance 1832 Excke, 1857 Dienger, 1858 Ritter, 1864 Chau- 

 VENET, 1867 Ha^^sen and Mekriman's Elements of the Method of 

 Least Squares (London, 1877, 8vo). 



The demonstration as given by Gauss contains three defects. ]. 

 It is not recognized that tlie probability of a definite error x is an 

 infinitesimal ; this is avoided by some later writers. 2. The dis- 

 tinction between true errors and residuals (or calculated errors) is not 

 sharply drawn ; according to Gauss's reasoning the law q)(x)=zcfr^^^^ 

 is not strictly a " law of facility of error" but only a law of distribu- 

 tion of residuals. 3. The rule of the aritlimetical mean is assumed. 

 For critical analyses of this proof see below, 1843 Reuschle 1844 

 Ellis and 1872 Glaisiier. 



Practical features of the method, — the formation of normal equa- 

 tions and the determination of weights and degrees of precision are 

 also discussed in the Section and hints are given regardino- its use in 

 astronomy. On page 221 is an attempt to justify the principle of 



