630 PROFESSOR BOOLE ON THE COMBINATION 
petitio principii. Tt 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 7 and r+0,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, @ prior?, 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 formule. No one 
has shown this more clearly than Mr Exuis. The precise value of Sir Joun HeEr- 
SCHEL’s principle, as corrected by him, | 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, 
