437 



reduced to a form in which its elements are logical. If a quadrant 

 of the celestial arc be taken as the unit of magnitude, then any 

 observed elevation of a star, regarded as a physical point, in a parti- 

 cular quadrant of elevation, will be the probability furnished by that 

 observation, that a pointer directed at random to the given quadrant 

 will point somewhere below the star. 



This is clear from the ordinary definition of the measure of proba- 

 bility. For, supposing the observed altitude to be 10°, then the 

 number of different directions that may be given to the pointer, 

 is to the number of such directions falling below the star, as 90 : 10, 

 eras 9 : 1. Hence the probability of the direction of the pointer 

 being below the star at J =the altitude of the star, considering the 

 quadrant as unity. 



In considering this problem, the author first makes some observa- 

 tions on the different principles which have been employed in its 

 discussion, viz. : the principle of the arithmetical mean regarded as 

 axiomatic in its nature, the principles of mechanical analogy, of geo- 

 metrical consistency, &c. And he suggests that the agreement of 

 results deduced in such various ways is to be taken as an evidence 

 of a certain harmony between the intellectual powers and the exter- 

 nal universe considered as their actual sphere of exercise. 



Reducing the problem, as above intimated, to logical elements, he 

 then applies to it the method developed in the " Laws of Thought." 

 The following, though not the most general, is the most interesting 

 result to which the application leads : — 



If n observations of the altitude of a star are to be combined, and 

 if Cp Cj) ■ • • <^" ^'■^ the several probabilities that these respective ob- 

 servations are absolutely correct, and if p^, p.^, . ./>„ are the altitudes 

 which they furnish, then the most probable altitude will be 



rrcrp.+n::c-/'--+rrF„^" 



i-«"i ■'"i-c., ■ ■■■ 1-C„ 

 This result is in accordance with the familiar form 



W,p, + Wjpj... +W„pn 



where W^ W., . . . W„ represent what are termed the weights of the 

 observations. But it is free from any admixture of mechanical ana- 

 logy, and it expresses the so-called weights as functions of certain 

 initial probabilities, viz. : the several probabilities of absolute cor- 

 rectness in the observations. Even if these probabilities are, as in 



