x.] THE THEORY OF PROBABILITY. 201 



manner in throwing it up, is almost sure to occasion a 

 slight preponderance in one direction. But as we do not 

 previously know in which way a preponderance will exist, 

 we have no reason for expecting head more than tail. Our 

 state of knowledge will be changed should we throw up 

 the coin many times and register the results. Every throw 

 gives us some slight information as to the probable 

 tendency of the coin, and in subsequent calculations we 

 must take this into account. In other cases experience 

 might show that we had been entirely mistaken ; we might 

 expect that a die would fall as often on each of the six 

 sides as on each other side in the long run ; trial might show 

 that the die was a loaded one, and falls most often on a 

 particular face. The theory would not have misled us : it 

 treated correctly the information we had, which is all that 

 any theory can do. 



It may be asked, as Mill asks, Why spend so much 

 trouble in calculating from imperfect data, when a little 

 trouble would enable us to render a conclusion certain by 

 actual trial ? Why calculate the probability of a measure- 

 ment being correct, when we can try whether it is correct ? 

 But I shall fully point out in later parts of this work that 

 in measurement we never can attain perfect coincidence. 

 Two measurements of the same base line in a survey may 

 show a difference of some inches, and there may be no 

 means of knowing which is the better result. A third 

 measurement would probably agree with neither. To 

 select any one of the measurements, would imply that 

 we knew it to be the most nearly correct one, which we 

 do not. In this state of ignorance, the only guide is the 

 theory of probability, which proves that in the long run 

 the mean of divergent results will come most nearly to 

 the truth. In all other scientific operations whatsoever, 

 perfect knowledge is impossible, and when we have ex- 

 hausted all our instrumental means in the attainment of 

 truth, there is a margin of error which can only be safely 

 treated by the principles of probability. 



The method which we employ in the theory consists in 

 calculating the number of all the cases or events concerning 

 which our knowledge is equal. If we have the slightest 

 reason for suspecting that one event is more likely to 

 occur than another, we should take this knowledge into 



