ON THE STATISTICAL VIEW OF NATURE. 577 



method of taking the arithmetical mean in determining 

 what figure to accept in a number of slightly differing 

 computations. Where more than one quantity is to be 

 determined for instance, where from a series of obser- 

 vations dotted on a chart the continuous curve which 

 marks the course of a planet or comet is to be deduced 

 the simple method of averaging cannot be applied. 

 Every set of three complete observations suffices, as 

 Gauss has shown, to determine the elements or con- 

 stants of an elliptical orbit. But astronomers try to 

 get as many observations as possible, and none of these 

 is a repetition of the same observation as, for in- 

 stance, are the repeated weighings of a substance in 

 chemistry, of the measurings of a length in surveying, 

 or the counting of a number in statistics : on the con- 

 trary, each is the independent ascertainment of definite 

 positions in a moving object. It is clear that the 

 method of averaging must be more general than the 

 common-sense method of taking the arithmetical mean, 

 but must where the latter is applicable coincide 

 with it. It has been shown that the following rule 

 answers this purpose. Fix the average constants or 

 elements so that the sum of the squares of the differ- 

 ences between the observed and calculated positions is a 

 minimum. In mathematical language this results in the 

 algebraical determination of the constants in an equation. 



Whereas the labours of Gauss and the school of 

 astronomers which he headed in Germany were mostly 

 occupied in the mathematical proof of this rule, and 

 in its applications in astronomical and geodetic com- 

 putations, the doctrine of probabilities acquired a larger 



VOL. II. 2 



