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, — being 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. 



Problem II. 



34. Required the pi^obability of an event z, 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 ofy being q. 



Here we are concerned with three events, x, y, and z. For convenience and 

 uniformity I shall, in the solution of the problem, speak of x and y as events, as 



