square root of nine, we immersed the wedge with its edge downward 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 interior 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; this 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. Supposing 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 lineal 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.