576 



SCIENTIFIC THOUGHT. 



18. 



Method of 

 Least 

 Squares. 

 Gauss. 



elaborate calculations in astronomy, geodesy, and in 

 various physical and statistical researches. 



Bound up with the theory of Error is the celebrated 

 method of least Squares, first used by Gauss in 1795, 

 published by Legendre in 1805 in his memoir ' On a 

 New Method of Determining the Orbit of a Comet,' and 

 elaborately discussed by Laplace, Gauss, and many sub- 

 sequent writers to this day. 1 It may be looked upon 

 as an extension or generalisation of the common-sense 



1 In addition to the references 

 given in the notes to pp. 120 and 

 183 of vol. i., I can now recommend 

 two excellent summary accounts 

 of the history and theory of the 

 method of least squares the one 

 in Prof. Czuber's ' Bericht,' quoted 

 above (pp. 150 to 224) ; the other 

 in Prof. Edgeworth's article on 

 "The Law of Error" in the Sup- 

 plement to the last edition of 

 the ' Ency. Brit.' (vol. xxviii., 

 1902, p. 280, &c.) Prof. Cleve- 

 land Abbe, in a " historical note 

 on the method of least squares " 

 ('American Journal of Mathe- 

 matics,' 1871), has drawn attention 

 to the fact, that already in 1808 

 Prof. R. Adrain of New Brunswick 

 had arrived at an expression for the 

 law of error identical with the 

 formula now generally accepted, 

 without knowing of Gauss's and 

 Legendre's researches. See a paper 

 by Prof. Glaisher in the 39th vol., 

 p. 75, of the ' Transactions of the 

 Royal Astronomical Society.' The 

 logical and mathematical assump- 

 tions upon which the method is 

 based have been submitted to re- 

 peated and very searching criti- 

 cisms, many rigid proofs having 

 been attempted, and every sub- 

 sequent writer having, seemingly, 

 succeeded in discovering flaws in 

 the logic of his predecessors. In 

 connection with another subject, 



I may have occasion to point 

 out how nearly all complicated 

 logical arguments have shown 

 similar weakness, and how, in many 

 cases, the conviction of the correct- 

 ness or usefulness of the argument 

 comes back to the self-evidence of 

 some common - sense assumption, 

 which cannot be proved, though it 

 may be universally accepted. Many 

 analysts have tried to prove the 

 correctness of the everyday process 

 of taking the arithmetical mean, 

 but have failed. Prof. Czuber says, 

 inter alia (loc. cit., p. 159): "The 

 fact that Gauss, in his first demon- 

 stration of the method of least 

 squares, conceded to the arithmeti- 

 cal mean a definite theoretical 

 value, has been the occasion for a 

 long series of investigations concern- 

 ing the subject, which frequently 

 showed the great acumen of their 

 authors. The purpose aimed at 

 viz., to show that the arithmetical 

 mean is the only result which ought 

 to be selected as possessing cogent 

 necessity, hereby giving a firm 

 support to the intended proofs, has 

 not been attained, because it cannot 

 be attained. Nevertheless, these 

 investigations have their worth be- 

 cause they afford clear insight into 

 the nature of all average values and 

 into the position which the arith- 

 metical average occupies among 

 them." 



