Fundamental Principles of Scientific Inquiry. 375 



at these moments its distances from the starting-point are 

 5, 20, 45,... centimetres. Then to state that at all the 

 instants measured the displacement is connected with the time 

 by the equation 



5as = f 2 (l) 



is a mere matter of description, and a purely logical 

 consequence of the experimental data. No principle of 

 generalization is yet required. But suppose we want to 

 know where the disk was 7 seconds from the start. True, 

 it may have been 9'8 cms. from the starting-point, but it is 

 perfectly clear that nothing in our present data will tell us 

 that it was. The body could be anywhere when t = 7 sees., 

 and yet be at the observed places at the observed instants. 

 Thus we cannot interpolate or extrapolate from observed 

 numerical data without some further principle; and it is 

 here that induction enters and enables us to say that the 

 equation (1) holds for instants other than the observed ones. 

 In practice a physical law that gives no information for 

 values of: the variables other than a definite finite number 

 is not of much use : we are almost certain to want information 

 for some other values than these. But it cannot give these 

 with certainty. In the above case, if we attempt to predict 

 the value of x when t is 100 sees., the equation (1) gives 

 # = 2000 cms. ; but it is always possible that something may 

 happen to the disk before it gets so far. In particular, the 

 sizes of laboratory inclined planes are usually such that 

 the disk will have been stopped by the end of its channel 

 before it has gone any large fraction of this distance. Such 

 inferences are only probable. We may perhaps be able to 

 assert, in the absence of other knowledge, that our data 

 make it probable that x will be 2000 cms. when t is 100 sees.; 

 but we cannot say definitely that it will be. This type of 

 inference necessarily involves the notion of probability. 

 Hence this notion must be one of our concepts, and its laws 

 must be among our propositions. We have shown in a 

 previous paper * that there are no advantages, and several 

 definite drawbacks, in treating probability as a derived 

 notion ; and accordingly we shall treat it as a primitive 

 concept with postulates of its own, which are accepted 

 a priori. The result obtained by such inference is never 

 of the form "]) is true," where p is a proposition verifiable 

 by observation : it is always of the form " the probability 

 of p, given our data, is so much." Considering how general 



* " On certain Aspects of the Theory of Probability," Phil. Mag. 

 vol. xxxviii. pp. 715-73J (19] 9). 



