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X. On the Method of the Least Squares, as employed in Deter- 

 mining the Figure of the Earth, from Experiments xvith the 

 Pendulum, as well as by the Measurements of Arcs. By W. 

 Galbraith, Esq. A.M. 



To the Editors of the Philosophical Magazine and Annals. 

 Gentlemen, 



TN numerous papers that have lately appeared on the de- 

 -*■ termination of the figure of the earth, from experiments 

 by the pendulum, as well as by the measurements of arcs, the 

 method of the least squares has been much employed, as best 

 calculated to arrive at the truth. The application of this me- 

 thod to such a purpose I think may perhaps be questioned. 

 On investigating the principles on which it is founded, it will 

 be observed that each individual observation is nearly of equal 

 authority. It is generally supposed that the errors are irre- 

 gular and fortuitous, and have no part constant, and that 

 each and every observation contributes its share of accuracy 

 to the genei'al conclusion. But it does not, I apprehend, guard 

 against the effects of a favourable or unfavourable situation 

 in any or sevei'al of the observations, by which a gi'eater or 

 less effect may be produced by the amount of error, whatever 

 it may be in magnitude, upon the final result. By combining, 

 as that method sometimes requires, observations made under 

 favourable circumstances with those in unfavourable, a result 

 may be produced less accurate than if a judicious selection of 

 decisive observations had been made, by which means the me- 

 thod of the least squares would have contributed by the manner 

 in which it must be employed to have vitiated the conclusion. 



On combining observations for various astronomical pur- 

 poses, there is often no criterion from which it can be con- 

 cluded that any one among a very considerable number has 

 the least advantage over any of the rest ; though on some oc- 

 casions it is quite well known that an observation made under 

 peculiar circumstances, may be more decisive than all the 

 others, or at least the greater part of them. 



Now this is precisely the case with regard to the compres- 

 sion of the earth ; to determine which, two measurements of 

 distant arcs are much more conclusive than a considerable 

 number near one another, or at least, whose distances are 

 not considerable. But when the method of the least squares 

 is employed, the whole are combined, both those in favourable 

 and in unfavourable situations ; so that the advantages of the 

 distant ones are in some degree counterbalanced by the dis- 

 advantages 



