336 REASONING. 



We may, if we please, call the proposition " Three 

 is two and one," a definition of the number three, and 

 assert that arithmetic, as it has been asserted that 

 geometry, is a science founded upon definitions. But 

 they are definitions in the geometrical sense, not the 

 logical ; asserting not the meaning of a term only, 

 but along with it an observed matter of fact. The 

 proposition, " A circle is a figure bounded by a line 

 which has all its. points equally distant from a point 

 within it," is called the definition of a circle ; but the 

 proposition from which so many consequences follow, 

 and which is really a first principle of geometry, is, 

 that figures answering to this description exist. And 

 thus we may call, " Three is two and one," a defini- 

 tion of three ; but the calculations which depend upon 

 that proposition do not follow from the definition 

 itself, but from an arithmetical theorem presupposed 

 in it, namely, that collections of objects exist, which 

 while they impress the senses thus, ( o> m ay be sepa- 

 rated into two parts, thus, c . This proposition 

 being granted, we term all such parcels Threes, after 

 which the enunciation of the above-mentioned phy- 

 sical fact will serve also for a definition of the word 

 Three. 



The Science of Number is thus no exception to the 

 conclusion we previously arrived at, that the processes 

 even of deductive sciences are altogether inductive, 

 and that their first principles are generalisations from 

 experience. It remains to be examined whether this 



general properties or its axioms to be of necessity inductively con- 

 cluded from the consideration of particular cases. And surely this 

 is the way in which children do acquire their knowledge of number, 

 and in which they learn its axioms. The apples and the marbles 

 are put in requisition, and through the multitude of gingerbread 

 nuts their ideas acquire clearness, precision, and generality." 



