288 REASONING. 



fundamental truths of that science all rest on the evidence of 

 sense ; they are proved by showing to our eyes and our fingers 

 that any given number of objects, ten balls for example, may 

 by separation and re- arrangement exhibit to our senses all the 

 different sets of numbers the sum of which is equal to ten. 

 All the improved methods of teaching arithmetic to children 

 proceed on a knowledge of this fact. All who wish to carry 

 the child's mind along with them in learning arithmetic ; all 

 who wish to teach numbers, and not mere ciphers now teach 

 it through the evidence of the senses, in the manner we have 

 described. 



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 on 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 aline 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 in geometry, is, that figures answering to this 

 description exist. And thus we may call " Three is two 

 and one" a definition of three; but the calculations which 

 depend on 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, , may be separated into two parts, 

 thus, oo o. This proposition being granted, we term all 

 such parcels Threes, after which the enunciation of the above 

 mentioned physical fact will serve also for a definition of the 

 word Three. 



The Science of Number is thus no exception to the conclu- 

 sion we previously arrived at, that the processes even of de- 

 ductive sciences are altogether inductive, and that their first 

 principles are generalizations from experience. It remains 

 to be examined whether this science resembles geometry in 

 the further circumstance, that some of its inductions are not 



