Cube Root Easily Found.
Take another example of inversion, this time in the field of mensuration. Every schoolboy knows that cubes respectively one, two, three, and four inches in diameter have contents respectively of one, eight, twenty-seven, and sixty-four cubic inches; that is, the contents vary as the cubes of the diameters of these solids. This is true of all solids alike in form. Cones, therefore, which have an angle of let us say fifteen degrees at the apex, vary in contents as the cube of their heights. Cones usually are looked at as they rest on their bases; it is worth while to consider them reversed, pointing downward. An inverted cone, duly supported on a frame allowing motion upward and downward, and dipping into a cylinder partly filled with water, is a simple means of extracting cube root within say one and ten as limits. The cone should be marked off into tenths, and the cylinder, between high and low-water, into thousandths. On a similar plan a tapering wedge acts as a square-root extractor, displacing water as the square of its depth of immersion.
Cube-root extractor.
The cone displaces water as the cube of its depth of immersion, in this case within 1 and 3 as limits.
Square-root extractor.
Wedge displaces water as the square of its depth of immersion.