236 DANIEL BERNOULLI. 



The following is all that Daniel BeiTioulli contributes to the 

 theory. Let m and n be lai-ge numbers ; let 



|2n 1 



u = 



V = 



2m 1 



in f 



He shews that approximately 



II /^m-\- 1 



V V 4w + 1 * 



/I IN"" 

 He also states the following : in the expansion of f ^ + „ I 



the jj}'^ term from the middle is approximately equal to —2 . 



These results are included in those of Stirling and De Moivre, 

 so that Daniel Bernoulli's memoir was useless when it appeared; 

 see Art. 837. 



424. The next memoir by Daniel Bernoulli is entitled Di- 

 judicatio maxime prohabilis plurium ohservationum discrepantitwi 

 Clique verisimillima inductio inde formanda. This memoir is con- 

 tained in the Acta Acad. ...Petrop. for 1777, pay^s piHor ; the 

 date of publication of the volume is 1778 : the memoir occupies 

 pages 8 — 23 of the part devoted to memoirs. 



425. The memoir is not the first which treated of the errors 

 of observations as a branch of the Theory of Probability, for 

 Thomas Simpson and Lagrange had already considered the sub- 

 ject ; see Art. 371. 



Daniel Bernoulli however does not seem to have been ac- 

 quainted with the researches of his predecessors. 



Daniel Bernoulli says that the common method of obtaining 

 a result from discordant observations, is to take the arithmetical 

 mean of the result. This amounts to supposing all the observa- 

 tions of equal weight. Daniel Bernoulli objects to this supposition, 

 and considers that small errors are more probable than large 

 errors. Let e denote an error ; he proposes to measure the pro- 

 bability of the error by ^(j-'^ — e^), where 7- is a constant. Then 



