J. F. ENCKE ON THE METHOD OF LEAST SQUARES. 329 



several observations have been made. Rightly understood, the 

 confidence which we place in all quantities derived by expe- 

 riment in any science, rests essentially on this proposition ; it 

 may, therefore, safely be affirmed concerning it, that its truth 

 has been confirmed by experience. The deduction here given 

 shows somewhat more clearly than would be done by the 

 simple statement of the proposition itself, the suppositions on 

 which it is founded. If the observations are made under strict- 

 ly equal circumstances, and if in two observations a positive 

 and a negative deviation of equal amount are regarded as equal, 

 the arithmetical mean is the only value which does not con- 

 tradict these suppositions. Then it cannot well be denied that 

 the same value ought to be obtained, whether the observa- 

 tions are considered all together, or divided into arbitrary 

 groups, provided only that no arbitrary supposition is made 

 in the combination of the results of these groups amongst them- 

 selves. To deny this, would be to deny that there is any value 

 which ought to be chosen in preference to others. It may 

 perhaps serve to illustrate the importance of the supposition 

 of the equality of the observations in reference to the arith- 

 metical mean, if we refer to the example furnished by Lam- 

 bert, in the Photometrie, §. 276, in which the arithmetical 

 mean obviously does not give the greatest approximation to 

 the truth. The periphery of a circle is always between the 

 values of the perimeter of an inscribed and a circumscribed 

 polygon of an equal number of sides. If, therefore, we consider- 

 ed the perimeter of a polygon of n sides as an observation of the 

 length of the periphery, and regarded the arithmetical mean be- 

 tween the inscribed and circumscribed polygon of n sides as the 

 most probable value of the periphery, we should be in error. 

 We come nearer the truth if we add to the perimeter of the in- 

 scribed polygon the third part of the difference between the two. 

 Whether, therefore, we regard the principle of the arith- 

 metical mean, in observations of equal worth, as a fundamental 

 proposition which experience has confirmed, — or whether we 

 prefer to take those propositions, on which the deduction here 

 given is based, as more simple fundamental propositions re- 

 quiring no proof, — in either case the founding of the application 

 of the calculus of probabilities to observations on the principle 

 of the arithmetical mean is, perhaps, of all the modes of demon- 

 stration, that which is most useful to the practical mathemati- 



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