REMAINING LAWS OF NATURE. 



401 



pebbles or of any other objects, we 

 put them together into the particular 

 sort of parcels or aggregates called 

 twelves ; and put together these 

 twelves again into similar collec- 

 tions ; and, finally, make up twelve 

 of these largest parcels : the aggre- 

 gate thus formed will be such a one 

 as we call 1728 ; namely, that which 

 (to take the most familiar of its modes 

 of formation) may be made by joining 

 the parcel called a thousand pebbles, 

 the parcel called seven hundred peb- 

 bles, the parcel called twenty pebbles, 

 and the parcel called eight pebbles. 



The converse proposition, that the 

 cube root of 1728 is 12, asserts that 

 this large aggregate may again be de- 

 composed into the twelve twelves of 

 twelves of pebbles which it consists of. 



The modes of formation of any 

 number are innumerable ; but when 

 we know one mode of formation of 

 each, all the rest may be determined 

 deductively. If we know that a is 

 formed from b and c, b from a and e, 

 c from d and /, and so forth, until we 

 have included all the numbers of any 

 scale we choose to select, (taking care 

 that for each number the mode of 

 formation be really a distinct one, 

 not bringing us round again to the 

 former numbers, but introducing a 

 new number,) we have a set of pro- 

 positions from which we may reason 

 to all the other modes of formation 

 of those numbers from one another. 

 Having established a chain of induc- 

 tive truths connecting together all the 

 numbers of the scale, we can ascer- 

 tain the formation of any one of those 

 numbers from any other by merely 

 travelling from one to the other along 

 the chain. Suppose that we know 

 only the following modes of forma- 

 tion : 6 = 4 + 2, 4 = 7-3, 7p5 + 2, 



5 = 9-4. We could determine how 



6 may be formed from 9. For 6 = 4 

 + 2 = 7-3 + 2 = 5 + 2-3 + 2 = 9-4-f- 

 2-3-1-2. It may therefore be formed 

 by taking away 4 and 3, and adding 2 

 and 2. If we know besides tliat 2 -l- 

 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 ascer- 

 taining all the rest. And since things 

 which are uniform, and therefore 

 simple, are most easily received and 

 retained by the understanding, there 

 is an obvious advantage in selecting 

 a mode of formation which shall be 

 alike for all ; in fixing the connotation 

 of names of number on one uniform 

 principle. The mode in which our ex- 

 isting 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 numbet 

 next below it in magnitude, 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 aggre- 

 gates each equal to one of the succes- 

 sive 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 com- 

 prehensive 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, obvi- 

 ous to the senses in all cases which 

 can be fairly referred to their decision, 

 and so general as to be co-extensive 

 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 applica- 

 tion of this law, or of other laws 

 capable of being deduced from it. 

 This is our warrant for all calcula- 

 tions. 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 

 2 C 



