OF TESTIMONIES OR JUDGMENTS. 631 
that it belongs to testimony, in its normal character, to be cumulative,—to judgment, 
to require the application, in some form or other, of the principle of means or 
averages ; but that all departures from these normal states involve the blending of 
the two elements together, in proportions determined by the degree of the de- 
flection. 
Now, although it does not belong to the theory of probabilities, in its formal 
and scientific character, to pronounce upon the material character of a pro- 
blem, and to say whether its data are in their own nature cumulative or not, yet 
the results to which the theory leads are, in a very remarkable degree, accordant 
with the distinctions which have just been pointed out. I shall show that the 
solution of the problem of the combination of testimonies, when the data are 
presented in a purely formal character, and without any adventitious principle, 
involves arbitrary constants, and is therefore indefinite,—heing capable, however, 
under certain circumstances, of assuming a definite form. I shall show that 
such a form is assumed when the circumstances are such as to give to the testi- 
monies the highest degree of cumulative character. I shall then solve the pro- 
blem a second time, introducing that adventitious principle which I have already 
exemplified in the problem of the reduction of astronomical observations, and 
which appears to me to contain the true theory of means or averages. The form 
of the solution thus obtained, which is also perfectly definite, will apply to the 
case, in which it is our object, not to combine testimonies, in the ordinary sense 
of the term, but to determine the mean of expectations founded upon the issues 
of conflicting judgments. To one point of importance I must again, before enter- 
ing upon the analytical investigation, ask the attention of the reader. It is, that 
in the present subject, the question of the right application of a formula is quite 
distinct from that of the validity of the processes by which that formula is de- 
rived from its data. The latter is a question of formal science, the former in- 
volves considerations which belong rather to the philosophy of the human mind. 
I will first express the problem which we have to consider in a general form, 
equally applicable to the combination of testimonies or of judgments. I shall 
consider the fact of a testimony having been borne, or an observation made, as a 
circumstance or event affecting our expectation of the event to which it has re-- 
ference. ; 
Prose II. 
34. Required the probability of an event 2, when two circumstances x and y are: 
known to be present,—the probability of the event z, when we only know of the exist- 
ence of the circumstance x being p,—and its probability when we only know of the 
existence of y being q. 
Here we are concerned with three events, z, y,and z. For convenience and 
uniformity I shall, in the solution of the problem, speak of # and y as events, as, 
