182 Mansfield Merrhnan — List of Writings relating 



The general term of the binomial {p-\-q)"\ in which p-\-q:=z\^ is 



shovvn to approach the form , as m indefinitely increases. 



-v 2 Ttmpq 

 See 1830 Poisson. 



1836 RouvROY. Ueber die Methode der kleinsten Quadrate. 

 Appendix to his Mechanih (Dresden and Leipzig, 8vo). 



1837 Hagen. Grimdzilge der WahrscJieinlichkeits-Reeluiung. 



Berlin, 8vo. — Second edition, see 1867. 



This work contains Hac^en's proof of the Method of Least Squares. 

 It is based upon the following hypothesis: "Der Beobaclitungsfehler 

 ist die algebraische Summe einer unendlich grossen Anzahl elemen- 

 tarer Fehler, die alle gleichen Werth haben und eben so leicht positiv, 

 wie negativ sein konnen." This postulated, the proof consists in 

 finding the general term of the expansion of (^ \- ^)""', m being 

 indefinitely large. The law of facility of error takes the form 



(p{,v) z= [7Tnt)~^e '" from which the principle of Least Squares at once 

 follows. 



The algebraic work of PL\gen's method had in a somewhat difierent 

 form been given by Laplace in the Theorie .... des Prohahilites 

 (1812), p. 301, and in the articles 1830 Poisson and 1836 Poisson. 

 Hagbn's method is more elementary and in connection with his orig- 

 inal hypothesis forms, I think, one of the best proofs of the Method 

 of Least Squares. 



Hagen's proof is given in the writings 1849 Wittstein, 1850 Encke, 

 1852 DiENGER, and in a modified form 1846 Quetelet, 1865 Tait, 

 1866 Natani and others. Also see Price's Integral Calculus (Ox- 

 ford, 1865), pp. 376-379. A discussion between Kummell and Mer- 

 RiMAN concerning this proof is now (Oct., 1877) going on in the flour. 

 Manklin Institute ; see Vol. CIV, pp. 173-187, 270-'274, et sq. 



1837 PoissoN. '■ Reeherehes sur la prohabilite des jxigemens en ma- 

 tih-e criminelle et en niatihre civile, precedees des rhgles g'enerales du 

 calcul des prohahilites? Paris, 4to, pp. ix, 415. — German trans, by 

 ScHNUSE called Lehrhuch der Wahrscheinlichkeitsrechnung ; Braun- 

 schweig, 1841, 8vo. 



The matter of Poissox's previous memoirs o)i the law of great 

 numbers is reproduced in Chap. Ill, and of those on the probability 

 of the mean in Chap. IV. 



1838 Bessel and Baeyer. ' Gradmesswig in, Ost-Prensseu. und 

 ihre Verhindung niit p)reussische und russische Dreiecksketten.'' Ber- 

 nn, 4to, p]). xiv, 452. 



A geodetic work of great value, containing many applications of 

 the Method of Least Squares. See extracts in Abhandl. von Bessel, 

 (Leipzig, 1875), Vol. Ill, pp. 82-138. 



