8o8 



POPULAR SCIENCE MONTHLY. 



first numbers. We take, a sheet of paper ruled in squares and shade 

 the first square of the first line, then the first two squares of the sec- 

 ond line, the first three of the third, etc. (Fig. 7). The whole num- 

 ber of squares shaded in this manner represents visibly the sum of the 

 first whole numbers up to any one we may choose to 7 in the figure. 

 If we give this paper to the child and ask him to return it, he will 



FIG. 7. 



very easily perceive that the figures formed by the white and the 

 black squares are alike. The number sought for will therefore be 

 equal to half the sum of the squares that is, in the present example 



1 + 2 + 3 + 4+5 + 6 + 7 = (7 X 8) : 2 = 28, 



we can prove by reasoning that if n be taken to represent the last 

 number we shall have for the sum 



o_ n (n + 1) 

 2 



I introduce this formula to define my thought better, but one can 

 make the child perceive the numbers that are wanted without writ- 

 ing down a single character. 



Somewhat similar is the method of finding the sum of the odd 

 numbers. For this it will be enough to take our square-ruled sheet 

 of paper and shade the first square on the left, then the three squares 

 around it, which will form with it a square (1 + 3 = 4); continu- 

 ing thus we obtain, as the figure readily shows (Fig. 8), a square 

 formed of a series of shaded zones, representing the series of odd 

 numbers, the examination of which will illustrate the property to 

 the child. 



In another direction it is possible to give the child algebraic ideas 

 much beyond anything we would imagine. Suppose, for example, 

 we want to give him a conception of addition. He easily realizes 



