to the Method of Least Squares. 183 



1838 Bessel. ' Untersuchungeu tiber die Wahrseheiiilichkeit der 

 Beobachtnngsfehler.' Astron. Nachr., Vol. XV, col. 369-404. —Ab- 

 handl. von Bessel (Leipzig, 18V5), Vol. II, pp. 372-391. 



" Ich werde namlicli die Eiitstehnngsart der Beobaolitungsfehler 

 aus ihren Vrsacheu, zuni Grande des Folgenden macheii. Weiin man 

 aufangs die Fehler einer gewissen BeoV)achtungsart also aus eiiter, 

 auf gegebene Art wirkenden Ursaehe hervorgeliend betrachtet, so 

 wird dadurch ihre jedesmalige Grosse x eine gegebene Function eines 

 Arguments*?, welches in derselben Art willkilhrlich ist, wie das Fallen 

 eines Wiirfels. Aus dem Aiisdrucke x =,/'^ kann aber der Ausdruck 

 (p{x) abgeleitet werden, . . . ." 



Bessel seems to use the word Ursaehe in the sense of a, som'ce of 

 error. His first investigation is of a case arising in the measurement 

 of angles, where the error x is related to the Ursaehe B by the law 

 x-=:a sin B, every value for B between the limits ziz-hrr being equally 

 possible. The law of facility of error he finds (p{x) = 7r~^{a^ —x^)~'^j 

 and the probable error is 2.568 times greater than by the Method of 

 Least Squares. An example where x =. aS^, which he shows may 

 actually arise, gives also disagreeing results. 



In the second part of the investigation we read: "Ich werde nun 

 die Wahrscheinlichkeit eines Fehlers untersuchen, welcher aus der 

 Zusammeiiwlrkung mehrer, von einander unabhangiger Ursachen 

 ensteht," each error so arising being considered as equally likely to 

 be ])Ositive or negative. The i-esult of the investigation is that the 

 law of facility of error approximates closely to the exponential form 

 qj{x) z=. ee~^' * , provided that " viele Ursachen zur Hervorbringen 

 des Beobachtungsfehlers zusarmnenvnrhen^'' and "dass unter den, ai;s 

 den einzelen Ursachen hervorgehenden mittlern Fehlern, keiner die 

 ubrigen betrachtlich libertrefle," and these conditions Bessel thinks, 

 are present in most observations. 



This memoir is very valuable as showing that the exponential law 

 of facility is not to be regarded as an d priori rule, free from excep- 

 tion, and as throwing new light on the condition under which it 

 exists. On the whole it may be considered as a new proof of that 

 law and hence of the Method of Least Squares. 



1838 BiENAYME. 'Mcmoire sur la ])robabilite des resultats moyens 

 des observations; demonstration directe de la regie de Laj'LAOE.' 

 M'etn. . .par divers savants, . . Acad. Paris, Vol. V, })p. 513-558. 



The rule of Laplace here meant is a method for finding the proba- 

 bility of the error of the mean. The opening pages contain some 

 interesting historical remarks, but the investigationitself is very long 

 and tedious and seems to be of little A^alue. 



1838 DeMorgan, '■Essay on Probabilities,'' London, 12mo. pp. 

 xviii, 306, xl. 



This popular book devotes a clia|)ter to methods of finding weights 

 and estimating probabilities of mean results. It contains tables of 

 the error functions. 



