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CHAPTER V. 

 OF DEMONSTRATION, AND NECESSARY TRUTHS. 



$ 1 . IF, as laid down in the two preceding chap- 

 ters, the foundation of all sciences, even deductive or 

 demonstrative sciences, is Induction ; if every step in 

 the ratiocinations even of geometry is an act of induc- 

 tion ; and if a train of reasoning is but bringing many 

 inductions to bear upon the same subject of inquiry, 

 and drawing a case within one induction by means of 

 another ; wherein lies the peculiar certainty always 

 ascribed to the sciences which are entirely, or almost 

 entirely, deductive ? Why are they called the Exact 

 Sciences? Why are mathematical certainty, and the 

 evidence of demonstration, common phrases to express 

 the very highest degree of assurance attainable by 

 reason ? Why are mathematics by almost all philo- 

 sophers,, and (by many) even those branches of natural 

 philosophy which, through the medium of mathe- 

 matics, have been converted into deductive sciences, 

 considered to be independent of the evidence of 

 experience and observation, and characterised as 

 systems of Necessary Truth ? 



The answer I conceive to be, that this character 

 of necessity, ascribed to the truths of mathematics, 

 and even (with some reservations to be hereafter 

 made) the peculiar certainty attributed to them, is 

 an illusion ; in order to sustain which, it is necessary 

 to suppose that those truths relate to, and express 

 the properties of, purely imaginary objects. It is 

 acknowledged that the conclusions of geometry are 

 deduced, partly at least, from the so-called Defini- 



