168 * INDUCTION. 



each is also considered as formed by the addition of 

 a number of units less than ten, and a number of 

 aggregates each equal to one of the successive powers 

 of ten : and this mode of its formation is expressed by 

 its spoken name, and by its numerical character. 



What renders arithmetic a deductive science, is 

 the fortunate applicability to it of a law so compre- 

 hensive as " The sums of equals are equals:" or (to 

 express the same principle in less familiar but more 

 characteristic language,) Whatever is made up of parts 

 is made up of the parts of those parts. This truth, 

 obvious to the senses in all cases which can be fairly 

 referred to their decision, and so general as to be 

 coextensive with nature itself, being true of all sorts of 

 phenomena (for all admit of being numbered,) must be 

 considered an inductive truth, or law of nature, of the 

 highest order. And every arithmetical operation is an 

 application of this law, or of other laws capable of being 

 deduced from it. This is our warrant for all calculations. 

 We believe that five and two are equal to seven, on the 

 evidence of this inductive law, combined with the 

 definitions of those numbers. We arrive at that 

 conclusion (as all know who remember how they first 

 learned it) by adding a single unit at a time: 5 + 1=6, 

 therefore 5+1 + 1 = 6 + 1 = 7: and again 2=1+1, 

 therefore 5+2=5 + 1+1 = 7. 



$ 6. Innumerable as are the true propositions 

 which can be formed concerning particular numbers, 

 no adequate conception could be gained, from these 

 alone, of the extent of the truths composing the 

 science of number. Such propositions as we have 

 spoken of are the least general of all numerical truths. 

 It is true that even these are coextensive with all 

 nature : the properties of the number four are true of 



