150 INDUCTION. 



taking away 4 and 3, and adding 2 and 2. If we know besides 

 that 2 + 2 = 4, we obtain 6 from 9 in a simpler mode, by 

 merely taking away 3. 



It is sufficient, therefore, to select one of the various 

 modes of formation of each number, as a means of ascertaining 

 all the rest. And since things which are uniform, and there- 

 fore simple, are most easily received and retained by the un- 

 derstanding, there is an obvious advantage in selecting a mode 

 of formation which shall be alike for ail ; in fixing the con- 

 notation of names of number on one uniform principle. The 

 mode in which our existing numerical nomenclature is con- 

 trived possesses this advantage, with the additional one, that 

 it happily conveys to the mind two of the modes of formation 

 of every number. Each number is considered as formed by 

 the addition of an unit to the number next below it in magni- 

 tude, and this mode of formation is conveyed by the place 

 which it occupies in the series. And 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 the type of a deductive science, 

 is the fortunate applicability to it of a law so comprehensive 

 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 pheno- 

 mena, (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 



