MEASUREMENTS 205 



has done this that he has a right to say that on any sub 

 sequent occasion, when the same fact recurs, the same 

 result will ensue. This is the most valuable of the modifica 

 tions which are contributed by experience to the instinctive 

 belief. With its assistance we are enabled to distinguish 

 causal propositions which are true, and weed out those 

 which are false. It compels us to suspend our judgement 

 in all cases where the likeness, or exact similarity, has not 

 been demonstrated for both terms ; when it has been, 

 a single instance, as in the case of chemistry, is sufficient for 

 a complete induction. Here we find the explanation of 

 Mill s difficulty. A single perfect instance is always suffi 

 cient for an induction. If we want more, it is only because 

 the individual instances are imperfectly defined, and we 

 have to eliminate differences. Inductions from the 

 enumeration and comparison of instances are nothing but 

 imperfect and temporary expedients in cases where our know 

 ledge of the subject-matter is not yet so exact as to enable 

 us to define our terms with sufficient accuracy. They are 

 especially useful in biology, where the little understood 

 principle of life contributes an irrational element to all our 

 calculations. Now the only means we have of making 

 this demonstration is by measurement. Exact similarity 

 is predicable only of ideas that are measurable, that is to 

 say, of ideas attributed to the external world. When we 

 say that two things are exactly alike, what we mean is that, 

 in respect to the characters compared, both correspond to the 

 same number of rhythmical beats. We have no other idea 

 of exactness. Mathematical exactness is attained only 

 when the beats or intervals are treated as pure abstractions, 

 that is to say, as neither in space nor in time. When, in 

 the case of concrete objects, we say that the likeness is not 

 exact, we mean that if the intervals were reduced, and the 

 rhythm quickened, the numerical coincidence would dis 

 appear, and one object be represented by a higher number 

 than the other. And this process of reduction cannot be 

 carried beyond the point when the rhythm would merge 

 into a continuum, and cease to be an implement of measure- 



