317 



Article X. 



On the Method of Least Squares. By J. F. Encke, Director 

 of the Astronomical Observatory at Berlin. 



[From the Aslronomisches Jahrhuch for 1834.] 



The frequent application of the method of least squares, or of 

 the calculus of probabilities, to the results of observations, in- 

 duces me to hope that a useful service may be rendered, by 

 giving as brief and elementary a view as is possible of the pro- 

 positions on vfhich this method is founded, — adding thereto 

 certain rules which I know from much experience to be most 

 convenient in practical application. With this design I have 

 drawn the present paper from the following sources : Gauss, 

 Theoria motus corporum cceleslium, Ub. ii. sect. 3 ; Disquisitio de 

 elementis ellipticis Palladis. Com. Gott.recen. vol. i. 1808-1811 ; 

 LiNDENAU and Bohnenberger, Zeitschrift fur Astronomie 

 und verwandte Wissenschaften, bd. i. pp. 185, et seq. ; Theoria 

 combinationis observationum erroribus minimis obnoxice. Com. 

 Gott. recen. 1821 and 1823, Parts I. and II. ; combined with re- 

 marks by Bessel in the Fundamenta Astronomia, pp. 18 and 

 116, and in his treatise on the Comet of Olbers. No proposition 

 of any importance is here put forward which is not taken from 

 the above-mentioned sources ; but the form of the demonstration 

 has occasionally been altered with the view of rendering it more 

 easy of comprehension. I have not thought it necessary to refer 

 to the particular places where the several propositions are to be 

 found. 



The classical labours of other mathematicians, especially 

 those of Laplace and Poisson, agree perfectly with those here 

 given, as respects the results : the form of representation and 

 the mode of deduction are different, chiefly because Laplace 

 confined himself to a strictly theoretical view of the subject, 

 and appears to have viewed but one amongst the many ap- 

 phcations of the calculus of probabilities. For the present ob- 

 ject, it has been thought preferable to follow the path pursued 

 by the two above-named astronomers, who combine the strictest 

 theorj^with the happiest practical application of theoretical truths; 



