436 



Before proceeding to the solution of these problems, the author 

 notices a distinction between problems (relating to probabilities) of 

 which the elements are logical, and problems of which the elements 

 cannot, without being subjected to some previous mental transfor- 

 mation, be regarded as logical. He describes as logical those pro- 

 blems in which the data are the probabilities of certain simple events, 

 and the qusesitum the probability of some compound event, whereof 

 the simple events are the elements combined in any way as happen- 

 ing or not happening. If, for example, the probabilities that the 

 individuals A, B, C, will continue in life for ten years, be p, q^ r, 

 given fractions, and the qusesitum be the probability of all the lives 

 continuing, or of their all failing, or of any two falling and one 

 continuing ; these he calls logical problems. In like manner, if the 

 data be the probabilities of such compound events, and the qusesitum 

 be the probability of some other compound, or of the simple events 

 compounded, these also are logical. This the author expresses 

 generally. 



Given prob. <p^ (x, y, z), prob. <p^ (x, y, z), &c. j find prob. ^p {x, y, z). 



It may be observed, that as an event must either happen or not 

 happen, these functions can consist only of factors of the form x and 

 \—x. He observes, — " I regard the elements of a problem relating 

 to probability as logical when the numerical data are the probabi- 

 lities, in the mathematical sense, i.e., measures of probability of 

 certain events, and when the object sought is the probability of some 

 other event whose connection with the former is either, — 1st, That it 

 is a compound event of which they are the simple components; or, 

 2d, That both it and they may be resolved into the same compo- 

 nents. He employs the term ' compound event' to denote, not 

 merely the conjunction of certain simple events, but any kind of 

 combination of them in the way of happening or not happening, 

 happening together, or in alternation ; or in any way which is ca- 

 pable of being expressed by the grammatical forms of language, 

 and especially by the adverb not and the conjunctions and^ either , 

 or, &c. 



According to this definition, the problem of the reduction or com- 

 bination of different simultaneously-observed altitudes of a star is 

 not in its direct presentation a logical one. The immediate data are 

 not probabilities of events, but measures of a physical magnitude. 

 The author, however, remarks that this problem is capable of being 



