400 



INDUCTION. 



asserted in the definition of a num- 

 ber is a physical fact. Each of the 

 numbers two, three, four, &c., 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 

 cannot 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 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 cognisance. And although 

 a hundred and two horses are not so 

 easily distinguished from a hundred 

 and three, as two horses ai:e from three 

 — though in most positions the senses 

 do not perceive any difference — yet 

 they may be so placed that a differ- 

 ence will be perceptible, or else we 

 should never have distinguished them, 

 and given them different names. 

 Weight is confessedly a physical pro- 

 perty of things ; yet small differences 

 between great weights are as imper- 

 ceptible to the senses in most situa- 

 tions, 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. 



What, then, is that which is con- 

 noted by a name of number? Of 

 course, some property belonging to 

 the agglomeration of things which 

 we call by the name ; and that pro- 

 perty is the characteristic manner in 

 which the agglomeration is made up 

 of, and may be separated into, parts. 

 I will endeavour to make this more 

 intelligible by a few explanations. 



When we call a collection of ob- 

 jects two, three, or four, they are not 

 two, three, or four in the abstract ; 

 they are two, three, or four things of 

 some particular kind ; pebbles, horses, 

 inches, pounds weight. What the 

 Hame of number connotes is the man- 



ner 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 two, 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 two, already 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 tivo may be united; 

 or one pebble may be added to an 

 aggregate of the kind called three. 

 Every succeeding number in the as- 

 cending series may be formed by the 

 junction of smaller numbers in a pro- 

 gressively greater variety of ways. 

 Even limiting the parts to two, the 

 number may be formed, and conse- 

 quently may be divided, in as many 

 different ways as there are numbers 

 smaller than itself ; and, if we admit 

 of threes, fours, &c., in a still greater 

 variety. Other modes of arriving at 

 the same aggregate present them- 

 selves, not by the union of smaller, 

 but by the dismemberment of larger 

 aggregates. Thus, three pebbles may 

 be formed by taking away one pebble 

 from an aggregate of four ; two pebbles, 

 by an equal division of a similar aggre- 

 gate, and so on. 



Every arithmetical proposition, 

 every statement of the result of an 

 arithmetical operation, is a state- 

 ment of one of the modes of forma- 

 tion of a given number. It affirms 

 that a certain aggregate might have 

 been formed by putting together cer- 

 tain other aggregates, or by with- 

 drawing certain portions of some 

 aggregate ; and that, by consequence, 

 we might reproduce those aggregates 

 from it by reversing the process. 



Thus, when we say that the cube 

 of 12 is 1728, what we affirm is this : 

 that if, having a sufficient number uf 



