The Nlmbek Concept. HI 



tion of number, it is necessary and sufficient to see, liear or 

 feel tliin<^s as difTcrentinted one from tlie other. The child 

 may, at the same time, be conscious of form, of the fact that 

 one body is larger or smaller than another, but /or?« and size 

 are not necessary for the iDrimitive number-concept. 



(8.) Counting is the simplest mathematical act. Foi- meas- 

 uring, the senses must supply to the mind more data than are 

 needed in counting, and the data must be more accurate. 

 For counting, the child only needs to see one object as sepa- 

 rate from another; for measuring, he must also see one as 

 greater than another. Take two rods, one twice as long as. 

 the other. For measurement, the child must not only see 

 them as distinct rods of unetjual length, but he must apply 

 the smaller rod to the larger, either in imagination or by 

 actual manipulation. The fact that the smaller rod can be 

 marked otf twice on the larger conveys no idea of ratio, unless 

 the child has, beforehand, the primitive nnmber-idea. If 

 this idea is present, then he may recognize the parts of the 

 longer rod as "two," and may obtain the idea of ratio. But 

 if the child has not the number-concept for '" two," then it 

 seems impossible for him to acquire an idea of the ratio be- 

 tween the lengths of the rods. The child must know that 

 the length of one rod is some number of times the length of 

 the other, before he can find out how many. If the primary 

 concept of number is a prere(iuisite to any attempt at meas- 

 urement, then one cannot find the origin of number in 

 measurement. 



Extension or the Number-Concept. (Ratio, Fhactions. ) 



"The first extension of the concept of number is the identification 

 of the ratio of any two magnitudes of the pame kind, and without quali- 

 tative distinction for the purposes of the comparison, as a number.''' 



"The measurement of any magnitude (concrete or abstract) is the 

 process of finding its ratio to another magnitude of the same kind, arbi- 

 trarily chosen as a unit. The measure of a magnitude is this ratio — a 

 number.''— .'\. Lefevke, Number and 7^s■ Algebra. 1890, pp. 61, l'J,j. 



