332 REASONING. 



seem to have come to one of Bacon's Prerogative 

 Instances ; an experimentum crucis on the nature of 

 reasoning itself. 



Nevertheless it will appear on consideration, that 

 this apparently so decisive instance is no instance at 

 all ; that there is in every step of an arithmetical or 

 algebraical calculation a real induction, a real infe- 

 rence of facts from facts ; and that what disguises the 

 induction is simply its comprehensive nature, and the 

 consequent extreme generality of the language. All 

 numbers must be numbers of something : there are 

 no such things as numbers in the abstract. Ten must 

 mean ten bodies, or ten sounds, or ten beatings of the 

 pulse. But though numbers must be numbers of 

 something, they may be numbers of anything* Pro- 

 positions, therefore, concerning numbers, have the 

 remarkable peculiarity that they are propositions con- 

 cerning all things whatever ; all objects, all existences 

 of every kind, known to our experience. All things 

 possess quantity ; consist of parts which can be num- 

 bered ; and in that character possess all the proper- 

 ties which are called properties of numbers. That 

 half of four is two must be true whatever the word 

 four represents, whether four men, four miles, or four 

 pounds weight. We need only conceive a thing 

 divided into four equal parts, (and all things may be 

 conceived as so divided,) to be able to predicate of it 

 every property of the number four, that is, every 

 arithmetical proposition in which the number four 

 stands on one side of the equation. Algebra extends 

 the generalization still farther : every number repre- 

 sents that particular number of all things without dis- 

 tinction, but every algebraical symbol does more, it 

 represents all numbers without distinction. As soon 

 as we conceive a thing divided into equal parts, with- 



