430 INDUCTION. 



ciple or premise of a science. The fact asserted in the definition of a num- 

 ber is a physical fact. Each of the numbers two, three, four, etc., denotes 

 physical phenomena, 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 can not be denied that two 

 apples are physically distinguishable from three apples, two horses from 

 one horse, and so forth ; that they are a different visible and tangible phe- 

 nomenon. 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 a hundred and two horses are not so easily distinguished 

 from a 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 confess- 

 edly a physical property of things; yet small differences between great 

 weights are as imperceptible to the senses in most situations, as small dif- 

 ferences 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. 



What, then, is that which is connoted by a name of number? Of 

 course, some property belonging to the agglomeration of things which we 

 call by the name ; and that property is, the characteristic manner in which 

 the agglomeration is made up of, and may be separated into, parts. I will 

 endeavor to make this more intelligible by a few explanations. 



When we call a collection of objects tioo, three, or four, they are not 

 two, three, or four in the abstract; they are two, three, or four things of 

 soma particular kind ; pebbles, horses, inches, pounds' weight. What the 

 name of number connotes is, the manner in which single objects of the 

 given kind must be put together, in order to produce that particular aggre- 

 gate. If the aggregate be of pebbles, and we call it Uoo, the name implies 

 that, to compose the aggregate, one pebble must be joined to one pebble. 

 If we call it three, one and one and one pebble must be brought together 

 to produce it, or else one pebble must be joined to an aggregate of the 

 kind called tico, ali-eady existing. The aggregate which we call four, has 

 a still greater number of characteristic modes of formation. One and one 

 and one and one pebble may be brought together; or two aggregates of the 

 kind called tiuo may be united; or one pebble may be added to an aggre- 

 gate of the kind called three. Every succeeding number in the ascending 

 series, may be formed by the junction of smaller numbers in a progressive- 

 ly greater variety of ways. Even limiting the parts to two, the number 

 may be formed, and consequently may be divided, in as many different 

 ways as there are numbers smaller than itself ; and, if we admit of threes, 

 fours, etc., in a still greater variety. Other modes of arriving at the same 

 aggregate present themselves, not by the union of smaller, but by the dis- 

 memberment of larger aggregates. Thus, three pebbles may be formed by 

 taking away one pebble from an aggregate of four ; ttco j^ebbles, by an 

 equal division of a similar aggregate ; and so on. 



Every arithmetical proposition ; every statement of the result of an 

 arithmetical operation ; is a statement of one of the modes of formation 

 of a given number. It affirms that a certain aggregate might have been 

 formed by putting together certain other aggregates, or by withdrawing 

 certain portions of some aggregate; and that, by consequence, we might 

 reproduce those aggregates from it, by reversing the process. 



