164 INDUCTION. 



with the definitions of the various numbers. Like 

 other so-called definitions, these are composed of two 

 things, the explanation of a name and the assertion of 

 a fact : of which the latter alone can form a first prin- 

 ciple or premiss of a science. The fact asserted in the 

 definition of a number is a physical fact. Each of the 

 numbers two, three, four, &c., denotes physical phe- 

 nomena, and connotes a physical property of those 

 phenomena. Two, for instance, denotes all pairs of 

 things, and twelve all dozens of things, connoting 

 what makes them pairs, or dozens ; and that which 

 makes them so is something physical ; since it cannot 

 be denied that two apples are physically distinguish- 

 able from three apples, two horses from one horse, 

 and so forth : that they are a different visible and 

 tangible phenomenon. I am not undertaking to say 

 what the difference is ; it is enough that there is a 

 difference of which the senses can take cognizance. 

 And although an hundred and two horses are not so 

 easily distinguished from an hundred and three, as 

 two horses are from three though in most positions 

 the senses do not perceive any difference yet they 

 may be so placed that a difference will be perceptible, 

 or else we should never have distinguished them, and 

 given them different names. Weight is confessedly a 

 physical property of things ; yet small differences 

 between great weights are as imperceptible to the 

 senses in most situations, as small differences between 

 great numbers ; and are only put in evidence by 

 placing the two objects in a peculiar position, namely, 

 in the opposite scales of a delicate balance. 



another. If A = a and A not = B, neither is a =B. For suppose 

 it to be equal. Then, since A = a and a = B, and since things 

 equal to the same thing are equal to one another, A = B ; which is 

 contrary to the hypothesis. 



