630 PROFESSOR BOOLE ON THE COMBINATION 



petitio principii. It is to be observed, that it is only the probability of deviation 

 from a fixed axis which follows, according to the above investigation, the law 

 expressed by Gauss's function. The probability of deviation in any direction to a 

 distance between r and r + 8 r from the mark, is expressed by a different function. 

 This would be fatal to any hypothesis which should represent Gauss's function as 

 determining, a priori, the actual law of deviation. There are indeed few cases in 

 which it can be determined what the law is, and writers on probability have been 

 far too anxious to interpret nature in accordance with their formulae. No one 

 has shown this more clearly than Mr Ellis. The precise value of Sir John Her- 

 schel's principle, as corrected by him, I conceive to be this, — that it establishes 

 an identity between the law of facility of error expressed by Gauss's function and 

 the law which in a special problem, involving the consideration of space and 

 motion, seems to accord with our most elementary conceptions of these things ; 

 and this identity I apprehend to be, not an accidental thing, but a very distinct 

 expression of that harmonious relation which binds together the different spheres 

 of thought and existence. 



33. We proceed next to the consideration of the second general problem, — that in 

 which it is proposed to determine the combined force of two testimonies or judg- 

 ments in support of a fact, the strength of each separate testimony being given. 



The problem has a material as well as a formal aspect. Thus oral testimonies 

 differ from the judgments which are furnished by the immediate personal obser- 

 vation of facts. And although no definite general laws have, so far as I am 

 aware, been assigned concerning the mode in which the material character of 

 the evidence affects expectation, it is not to be doubted that an influence does 

 proceed from this source. As respects testimony alone, there are cases in which 

 we feel that it is cumulative,— there are cases in which we feel that it is not 

 so ; and this difference we also feel depends upon the nature of the testimony 

 itself. But in the majority of cases, we should probably feel that the elements 

 upon which this difference of character depends are blended together, some 

 decided preponderance being due to the one or to the other. Testimony will 

 be chiefly or entirely cumulative which is given quite independently by different 

 persons, and is at the same time based upon different grounds. In proportion 

 as these conditions fail of being satisfied, the testimony partakes less and less 

 of the cumulative character. Still this possession of cumulative character may 

 be regarded as the standard by which the distinctive qualities of testimonies, as 

 affecting belief or expectation, may be estimated. In judgments founded upon the 

 personal observation of facts, though this character may be observed, the standard 

 seems to be different. When different modes of considering a subject — different 

 courses of experiment or inquiry — lead to different probabilities of a fact, some 

 making it more probable, some less, we generally feel that a kind of mean ought 

 to be taken among them. Perhaps the most succinct general statement would be, 



