REMAINING LAWS OF NATURE. 167 



included all the numbers of any scale we chuse to 

 select, (taking care that for each number the mode of 

 formation is really a distinct one, not bringing us 

 round again to the former numbers, but introducing a 

 new number,) we have a set of propositions from 

 which we may reason to all the other modes of for- 

 mation of those numbers from one another. Having 

 established a chain of inductive truths connecting 

 together all the numbers of the scale, we can ascertain 

 the formation of any one of those numbers from any 

 other by merely travelling from the one to the other 

 along the chain. Suppose that we knew only the 

 following modes of formation: 6 = 4 + 2, 4=7- 3, 

 7=5 + 2,5=9-4. We could determine how 6 may be 

 formedfromQ. For 6 = 4 + 2= 7-3 + 2 = 5 + 2-3 

 + 2=9-4 + 2-3 + 2. It may therefore be formed 

 by 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 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 prin- 

 ciple. The mode in which our existing numerical 

 nomenclature is contrived 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 

 magnitude, and this mode of formation is conveyed 

 by the place which it occupies in the series. And 



