174 31 a7is field Merrhnan — List of Writings relating 



1822 Encke. ' Die Entferming der Sonne von der Erde aiis dem 

 Venusdurchgrwge von 1761.' Gotha, 8vo, pp. 179. — Part II entitled 

 ' Der Venusdnrchgang von 1769 . . . .' Gotha, 1824, 8vo, pp. 112, 



The Method of Least Squares is applied to the reduction of 149 

 observations from the transit of 1761 and 106 from that of 1769, and 

 to the determination of probable errors. The most probal)le distance 

 of the sun from the earth is found to be 20666800 German geograph- 

 ical miles with the probable error of 89150 miles. 



1822 Gauss. 'Anwendung der Wahrscheinlichkeitsrechnung auf 

 eine Aufgabe der practischen Geometric.' Astronomische Nach- 

 richten^ Vol. I, col. 81-88. — See 1855 Bertrand. 



The problem is : To determine the position of a point from hori- 

 zontal angles taken at that point between other points whose position 

 is exactly known. A numerical example is given in which the nixm- 

 ber of known points is five and the number of angles is six. This is 

 often called Pothenot's problem; see 1840 Gerling, 1866 Schott. 



1823 Gauss. ' Theoria combinationis observationum erroribus 

 minimis obnoxijc.' (Jonirnent. Soc. Gottingen., Vol. V, pp. 33-90. 

 — Also Gauss Werke, Vol. TV (Gottingen, 1873, 4to), pp. 1-53. 

 — French trans, see 1855 Bertrand. 



This memoir contains Gauss's second Proof of the Method of Least 

 Squares. The following quotation from pages 37-38 sliows the hy- 

 pothesis upon which the proof is leased : " . . . . integral eyit'a; ^(a-) dx 

 ab ic =: — 00 usque ad a; = -|- o) extensum (sen valor medius quadrali 

 x^) aptissimum videter ad incertitudinem observationum in genere 

 definiendam et dimetiendam, ita ut e duobus observationum systema- 

 tibus, quoB quoad errorum fiicilitatem inter se differunt, ex prsecisione 

 prestare censeantiir, in quibus integraleyira? (p[a-) dx valorem minorem 

 obtinet." Gauss does indeed recognize and ])oint out that this is 

 only an arbitrary convention, but he justifies himself in adopting it 

 on the ground that the definition of most advantageous results must 

 be arbitrary, since the question is in its very nature indefinite, and 

 that his definition leads to simple operations. The values of the un- 

 known quantities found by his method he calls " valores maxime 

 plausibiles." 



Gauss's method leads to the rule of Least Squares, whatever be 

 the number of observations or whatever be the law of facility of 

 error provided only that positive and negative errors are eqixally 

 probable. For analyses of his proof see 1844 Ellis and 1872 

 Glaisher, the former regarding it as valid and the latter as unsatis- 

 factory. In my opinion it is but little more than a begging of the 

 question to assume that the mean of the squares of the errors is a 

 measure of precision. See below 1825 Ivory, 1847 Galloway and 

 1872 Helmert. 



The memoir contains an extended presentation of the practical fea- 

 tures of the method and in this respect is of great value. The algo- 



