REMAINING LAWS OF NATURE. 307 



6 = 9 — 4. We could determine how 6 may be formed from 9. For 

 6 = 4 + 2=7 — 3 + 2=5 4-2 — 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 forma- 

 tion of each number, as a means of ascertaining all the rest. And 

 since things which are uniform, and therefoi-e simple, are most easi^ 

 received and retained by the understanding, there is an obvious ad- 

 vantage in selecting a mode of formation which shall be alike for all ; 

 in fixing the connotation of names of number on one unifonn principle. 

 The mode in which our existing numerical nomenclature is contrived 

 possesses this advantage, with the additional one, that it happily con- 

 veys to the mind two of the modes of formation of every number. 

 Each number is cojisidercd as formed by the addition of an unit to the 

 number next below it in magnitude, and this mode of formation is con- 

 veyed by the place which it occupies in the sei'ics. 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 ai'ithmetic a deductive science, is the fortunate appli- 

 cability 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 phenomena (for 

 all admit of being numbered), must be considered an inductive truth, 

 or law of nature, of the highest order. And every arithmetical opera- 

 tion is an application of this law, or of other laws capable of being de- 

 duced 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 iinit 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 mimbers, 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 gen- 

 eral of all numerical truths. It is true that even these are coextensive 

 ^vith all nature : the properties of the number four are true of all ob- 

 jects that are divisible into four equal parts, and all objects are either 

 actually or ideally so divisible. But the propositions which com])ose 

 the science of algebra are true, not of a particular number, but of all 

 numbers ; not of all things under the condition of being divided in 

 ' a particular way, but of all things under the condition of being divided 

 in any way — of being designated by a number at all. 



Since it is impossible for different numbers to have any of their 

 modes of formation completely in common, it looks like a paradox to 

 say, that all propositions which can be made concerning numbers relate 



