46 THE POPULAR SCIENCE MONTHLY. 



square root of nine, we immersed the wedge with its edge down- 

 ward until it had displaced water to line nine on the jar's side. 

 On the wedge the water stood at line three, the square root of nine. 

 In a similar way the cone was observed to displace water as the 

 cube of its depth of immersion, and therefore could be impressed 

 into the service of extracting the cube root. For this purpose its 

 total play of displacement in a jar of five and a half inches in- 

 terior diameter was divided into twenty-seven parts, and the cone 

 was marked off into three sections. To find the cube root of eight, 

 we lowered the cone apex downward, until the water-level was 

 brought to eight on the jar's side ; at that moment the liquid 

 encircled the cone at section two, denoting the cube root of eight. 

 The pyramid immersed in the larger jar acted equally well as a 

 cube-root extractor. Measuring both cone and pyramid at each 

 of their sectional divisions, the boys were required to ascertain 

 the rule governing their increase of sectional area, and arrived at 

 the old familiar law of squares a law true not only of all solids 

 converging regularly to a point, but of all forces divergent or 

 radiant from a center, simply because it is a law of space through 

 which such forces exert themselves. 



While I was glad to use examples and models to instruct my 

 pupils, I wished them to grasp certain geometrical relations 

 through exercise of imagination. They had long known that the 

 area of a parallelogram is the product of its base and height ; they 

 were now required to conceive that any triangle has half the area 

 of a parallelogram of equal height and base. It was easy then to 

 show them the very old way of ascertaining the area of a circle, the 

 method which conceives it to be made up of an indefinitely great 

 number of triangles whose bases become the circle's circumference, 

 and whose altitude is the circle's radius. Rolling the cylindrical 

 model once around on a sheet of paper, its circuit was marked off ; 

 Hi is was made the base-line of a parallelogram having a height 

 equal to half the cylinder's breadth; half that area was clearly 

 equal to the surface of the circle forming the cylinder's section. 

 Another method of proving the relation between the area of a 

 circle and its circumference was followed by the boys with fair 

 promptness. I asked them to imagine a circular disk to be made 

 up by the contact of a great number of concentric rings. Sup- 

 posing the disk to be a foot in diameter and each ring to be the 

 millionth of a foot wide, I inquired, "How many rings would there 

 be ?" " Half as many, half a million." To the question, " What 

 would be the size of the average ring's circumference ?" " Half 

 that of the whole circle." was the reply. They were thus brought 

 to it that if a circle rolled around once is found to have 3*1416 lin- 

 eal units for its circumference, its area must be '7854, or one half of 

 one half as much, expressed in superficial units of the same order. 



