to the MeAhod of Least fSquares. 211 



Adrain's first proof is examined at length and its reasoning shown 

 to l)e defective. Then are analysed in order: 1. Gauss''s first proof, 

 including Encke's, DeMohgan's and Ellis's remarks on the arith- 

 metical mean ; 2. Laplace's method, Poisson's and Ellis's simplifi- 

 cations and Ivory's criticisms; 3. Gauss's second demonstration; 



4. Herschel's proof, with Ellis's and Boole's criticisms thereon ; 



5. Tait's and similar proofs ; 6. Donkin's proof of 1857. By means 

 of the index at the end of this list the reader may refer back to these 

 papers, where I have often quoted Glaisheu's remarks. 



It is considered unproved that the arithmetical mean gives the 

 most probable result. Gauss's second proof is regarded as resting 

 upon an arbitrary assumption, which practically assumes the point 

 to be proved. Laplace's method is considered as giving the only 

 correct and philosophical analysis of the question, and this Glaisher 

 shows leads directly to the exponential law of facility, provided that 

 the sources of error are very great in number and that positive and 

 negative errors are equally likely. " Tait's proof " is found insutti- 

 cient. The proofs of 1837 Hagen, 1838 Bessel, 1844 Donkin and 

 1870 Crofton are not discussed. 



Peirce's criterion for the rejection of doubtful observations is re- 

 garded as " destitute of scientific precision." '' . . . .under no circum- 

 stances have we a right to say an observation has no weight, though 

 it may be better to give it none than to give it as much as the best." 

 The method of assigning weights in such cases is hinted at ; see be- 

 low 1873 Glaisher. 



For accounts of the contents of the memoir see Monthly Notices, 

 Vol. XXXII, p. 241, ^ndJahrh. FoHschr. Math., Vol. IV, p. 92. 



1872 Helmert. ''Die AusgleichungsrecJuKDKj nach der Methode 

 der kleinsten Quadrate mit A}iwendimgen auf die Geodasie und die 

 Theorie der Messinstrmnente^ Leipzig, Svo, pp. xi, 348. 



The exponential law is regarded as a law proved by experience. 

 The arithmetical mean is said to be the most plausible value. Both 

 the first and second proofs of Gauss are given, and the second is re- 

 garded as better and more general. 



While the theoretical part of the book is not satisiactory, the 

 practical part renders it valuable for geodetic engineers. Condi- 

 tioned observations in particular are well treated. 



1872 HiLGARi). ' An a])plication of an Exponential Function.' 

 Proc. Amer. Assoc, for 1871, pp. 61-63. 



A certain statute relatino- to errors in coinage is discussed. 



1872 HoPKiNSON. On the calculation of empirical foi'muUe. 

 Messenger Math., Vol. II, pp. 65-67. 



A method less accurate than Least Squares. 



Trans. Conn. Acad., Vol. IV. 28 Oct., 18V7. 



