DEMONSTRATION, AND NECESSARY TRUTHS. 285 



tersecting other lines, forming an angle with one another, and 

 the like ; but not so a and b. These may represent lines or 

 any other magnitudes, but those magnitudes are never thought 

 of; nothing is realized in our imagination but a and b. The 

 ideas which, on the particular occasion, they happen to repre- 

 sent, are banished from the mind during every intermediate 

 part of the process, between the beginning, when the premises 

 are translated from things into signs, and the end, when the 

 conclusion is translated back from signs into things. Nothing, 

 then, being in the reasoner's mind but the symbols, what can 

 seem more inadmissible than to contend that the reasoning pro- 

 cess has to do with anything more ? We 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 inference 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 any- 

 thing. Propositions, therefore, concerning numbers, have the 

 remarkable peculiarity that they are propositions concerning 

 all things whatever ; all objects, all existences of every kind, 

 known to our experience. All things possess quantity ; con- 

 sist of parts which can be numbered ; and in that character 

 possess all the properties which are called properties of numbers. 

 That half of four is two, must be true whatever the word four 

 represents, whether four hours, 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 represents that par- 



