to the Method of Least Squares. 169 



Daniel Bernoulli's method agrees closely with modern theory. 

 See IVZH EuLER, 1785 Bernoulli and Toohunter, History of Prob- 

 ability, p. 236. 



1778 EuLER. ' Observationes in praecedentem dissertationem II- 

 lustr. Bernoulli.' Aeta Acad. Petrop. for 1777, Part I, pp. 24-33 

 of the memoirs. 



EuLER considers that Daniel Bernoulli is correct in objecting to 

 the arithmetical mean when the observations are of unequal pre- 

 cision, but that he " was quite arbitrary in proposing to make the 

 product of the probabilities a maximum. Euler proposes another 

 method which amounts to making the sum of the fourth powers of 

 the probabilities a maximum." — Todhunter, Hist, of Probability, 

 p. 237. 



1781 Laplace. 'Memoire sur les Probabilites.' Hist. Acad. 

 Paris for 1778, pp. 227-332 of the math. part. 



Pages 322-332 are devoted to the discussion of the mean to be 

 taken between discordant observations. Laplace says that by the 

 expression mean or mean result an infinite number of things may be 

 understood. Of these the one which is implied in adjusting observa- 

 tions is a value such that the resulting error shall be a minimum, and 

 this corresponds to a value such that the sum of the errors, each mul- 

 tiplied by its probability shall be a minimum. The method of his 

 memoir of 1774 is then substantially repeated. 



Laplace remarks that if dr« be the limits of error, the law of 



\ a 



facility ought to be q}{;x)=z — log^-. His methods lead to unmanag- 



able equations. 



2a 



1785 Bernoulli (Jean, III). 'Milieu.' Art. m the Encyclo2)edie 

 M'ethodique (Paris, 4to), Yol. II, pp. 404-409. — Second edition, 

 1789, entitled DicMonaire Encycl. des Mathematiques. 



Boscovicii's method of 1760 and Lambert's of 1765 are referred 

 to, and an account is given of Daniel Bernoulli's memoir of 1778, 

 which differs slightly from the memoir itself. An account of La- 

 grange's memoir of 1774 is presented with considerable fullness. 

 See Todhunter, Hist, of Pr oh ability, p. 442. 



1788 Euler. ' Eclaircissemens sur le memoire de Mr De La 

 Grange insere dans le V'= volume de Melanges de Turin, concernant 

 la methode de prendre le milieu entre les resultats de plusieurs obser- 

 vations, &c.' Nova Acta Acad. Petrop. for 1785, Vol. Ill, pp. 289- 

 297 of the memoirs. 



The memoir seems to have no value; See Todhunter, Hist, of 

 Probability, p. 250. 



