DEMONSTRATION, AND NECESSARY TRUTHS. 167 



This, however, though it looks so plausible, will not stand examin- 

 ation. The expression, " two pebbles and one pebble," and the ex- 

 pression, " tlirce pebbles," stand indeed for the same aggregation of 

 objects, but they by no means stand for the same physical fact. They 

 are names of the same objects, but of those objects in two diflerent 

 stales : though they <^('note the same things, their fo«notation is differ- 

 ent. Three pebbles in two separate parcels, and three pebbles in 

 one parcel, do not make thu same impression on our senses ; and the as- 

 sertion that tlie very same pebbles may by an alteration of place and by 

 arrangement be made to pr<)duce either the one set of sensations or the 

 other, though it is a very familiar proposition, is not an identical one. 

 It is a truth kuo\\ni to us by early and constant experience : an induc- 

 tive trutli : and such truths are the foundation of the science of Num- 

 ber. The fundamental truths of that science all rest upon 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 rearrangement exhibit to our senses all the different sets of num- 

 bers riie sum of which is equal to ten. All the improved methods of 

 teaching arithmetic to children proceed upon a knowledge of this fact. 

 All who wish to cany the child's mind along with tliem in learning 

 arithmetic; all who (as Dr. Biber in his remarkable Lectures on Edu- 

 cation, expresses it) 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 geometiy, is a science founded upon definitions. But 

 they are definitions in the geometrical sense, not the logical ; asseiting 

 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 conse- 

 quences 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 definition of three ; but the calcula- 

 tions which depend upon that proposition do not follow fi-om the defi- 

 nition 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, o o o. This propo- 

 sition 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 conclusion we 

 previously annved at, that the ju'ocesses even of deductive sciences are 

 altogether inductive, and that their first principles are generalizations 



* See, for illustrations of various sorts, Professor Leslie's Philosophy of Arithmetic ; and 

 see also two of the most efficient books ever written for training the infant intellect, 

 Mr. Horace Grant's Arithmetic for Young Children, and his Secotid Utage of Arithmetic, 

 both published by the Society for the Diffusion of Useful Knowledge. 



"Number," says the reviewer of Mr. Whevvell, already cited, " we cannot help regarding 

 as an abstraction, and consequently its general properties or its axioms to be of necessity 

 inductively concluded 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." 



