168 Mansfield Merrhnan — Ziist of Writings relating 



Thus if three measurements of an angle give 



J/, = «°&'0", 3f^ = a°b'4.0" and M^ = a°h'50", 



we have ^>=:40", 5^=10" and §'==0.25/); then from the table ,t:=37" 

 and the adjusted result is a° b' 3^", while by the usual rule of the 

 average we would have a° b' 30". By Laplace's table the mean 

 will lie nearer to the two observations which most nearly agree, than 

 in the common method. 



For remarks on this memoir see Todhunter, History of Proba- 

 bility^ p. 469, and Glai.shek, Me^n. Astron. Soc. Loncl.^ 1872, Vol. 

 XXXIX, pp.'^121-123. 



[1774] Lagrange. 'Memoire sur I'utilite de la mi'thode de pren- 

 dre le milieu les resultats de plusieurs observations ; dans lequel on 

 examine les avantages de cette methode par le calcul des proba- 

 bilites; & oti I'on resoud difterons problemes relatifs a cette matiere.' 

 Miscellanea Taurinetisia [Mel. Soc. Tarin) for 1770-1773, Vol. V, 

 pp. 167-232 of the math. part. 



This memoir is a more thorough presentation of the subject treated 

 by Simpson in 1757 with much new matter added. Lagrange 

 makes no allusion however to ])revious writings on the subject. The 

 expression " Law of Facility of Error" occurs here for the first time. 



For an extended account of the contents of the memoir see Tod- 

 hunter, History of Probability, pp. 301-313. See also below 1785 

 Bernoulli, 1788 Euler, and 1804 Tremblet. In 1850 Encke gives 

 a translation of part of the memoir, with comments. 



1778 Bernoulli (Daniel). ' Dijudicatio maxime probabilis plu- 

 rium observationum discrepantiimi atque verisimillima inductio inde 

 formanda.' Acta Acad. Fetrop. for 1777, Ft. I, pp. 3-23 of the 

 memoirs. 



The now familiar illustration of a marksman firing at a target is 

 here introduced, and the conclusion drawn that small errors are more 

 probable than large ones, and that the method of taking the arith- 

 metical mean " non sine ratione dubitare potest," since it supposes 

 the observations of equal weight. 



Daniel Bernoulli takes a circle yrrVr^ — x^ as representing the 

 law of facility of error, y being proportional to the probability of the 

 error cc, and r a constant. Then if observations give the errors cc,, 



X2, x^ , the product *>/r^—x^^ sfr^ — x^^ s/r^ — x^"^ must be a 



maximum to give the most probable value of the observed quantity. 

 He finds that "this value coincides with that given by the rule of the 

 arithmetical mean for one and for two observations, and that it 

 nearly coincides for three when a suitable value is given to r. For a 

 greater number than three his method leads to unmanageable equa- 

 tions. He closes by remarking that the problem is indeterminate. 

 See Zach's Monatliclie Correspondenz, 1805, Vol. XI, pp. 486-490. 



