Page:Popular Science Monthly Volume 38.djvu/55

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45
MY CLASS IN GEOMETRY.

of a cube having a base five inches square, and a wedge and pyramid of similar base and height. The second series comprised a cylinder, sphere, and cone, each five inches broad and high. Taking the first series, a moment's comparison of the sides of wedge and cube told that one contained half as much wood as the other; but that the pyramid contained a third as much as the cube was not evident. Weighing the pyramid and cube brought out their relation, but a more satisfactory demonstration was desirable, for what was to assure us that the two solids were of the same specific

PSM V38 D055 Demonstrating volumes of geomterical objects.jpg
Fig. 1. Fig. 2.

gravity? Taking a clear glass jar of an accurately cylindrical interior, measuring seven and a half inches in width by ten in height, it was half filled with water, and a foot-rule was vertically attached to its side. The models, which were neatly varnished, and therefore impervious to water, were then successively immersed and their displacement of the water noted. This proved that the pyramid had a third the contents of the cube, that the same proportion subsisted between the cone and cylinder, and that the sphere had twice the contents of the cone. Dividing the wedge by ten parallel lines an equal distance apart, I asked how the area of the smallest triangle so laid off, and that of the next smallest, compared with the area of the large triangle formed by the whole side of the wedge. "As the square of their sides" was the answer. Dipping the wedge below the surface of the water in the jar, edge downward, it was observed to displace water as the square of its depth of immersion. Reversing the process, the wedge became a simple means of extracting the square root. Dividing the vertical play of its displacement into sixteen parts drawn along the jar's side, we divided the wedge into four parts by equidistant parallel lines. Then, for example, if we sought the