to the Method of Least Squares. 169 . 



Grosse za begehen ; ihre Mittheiliing muss ich bis auf eine aiidere 

 Gelegenheit vei'sparen." See 18 10 Bessel, and also page 196 of the 

 memoir 181(3 Gauss. 



1815 Laplace. ' Sur I'application dii calcul des probabilites a la 

 philosophie iiaturelle.' Co?inaissance des Terns for 1818, pp. 361- 

 381. — First Supplement to third edition of Theorie . . . . des Proh. 

 (Paris, 1820, 4to), pp. 3-26. 



This is devoted partly to a general description of Lapi,ace's proof 

 of the Method of Least Squares, and partly to the discussion of the 

 probability of results obtained by that method ; a numerical exam- 

 ple illustrates the use of his formulae. See Todhuxtek, Hist, of 

 Prohahility., p. 610; also see 1869 Todhunter. 



1816 Beeck-Calkoen. Over de Theorie der Gemiddelde Waardij. 

 Verhandl. Nederland. Inst,., Vol. II, pp. 1-19. 



Treats of Laplace's method of adjustment of 1792. 



1816 Bessel, ' Untersuchungen ilber die Bahn des OLBERSSchen 



Kometen.' Abhandl. Akad. Berlin for 1812-13, pp. 117-160 of the 



math, section. 



Bessel defines the probable error as follows : " Ich verstehe unter 

 dieser Benennung die Grenze, die eine Anzahl kleinerer Fehler von 

 einer gleichen Anzahl grosserer trennt, so dass es wahrscheinlicher 

 ist, eine Beobachtung iiinerhalb jeder loeiteren Grenze von der Wahr- 

 heit abirren zu sehen, als ausserhalb derselben." If we designate by 



- — the mean of the errors all taken positively, by — — the mean of 



the squares of those errors, and by r the probable error of a single 

 observation, his demonstration shows that 



r=: 0.8453 — or ;•= 0.6745 



■i 



Bessel does not distinguish between true errors and residuals. 

 These formuhe he uses in finding tlie probable errors of the elements 

 of the orbits, which are deduced by the help of the Method of Least 

 Squares. 



1816 Gauss. ' Bestimmung der Genauigkeit der Beobachtungen.' 

 Zeitschr. f. AstroH. u. ver.Wiss.,Yo\. I, pp. 185-196. — Also Gauss 

 Werke, Vol. IV (Gottingen, 1873, 4to), pp. 109-117. —See 1855 

 Bertkand. 



This memoir gives three methods for finding the probable error 

 from given observations. The first, which is that usually presented 

 in text-books, finds that r the probable error of an observation of the 

 weight unity, is given (most probably) by 



Trans. Conn. Acad., Vol. IV. 22 Oct., 1877. 



