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. [Illustration: Cube-root extractor. The cone displaces water as the cube of its depth of immersion, in this case within 1 and 3 as limits.] [Illustration: Square-root extractor. Wedge displaces water as the square of its depth of immersion.] From Effect to Cause. A mechanic, no less than a geometer, may show sagacity in taking up a question in reverse, and reasoning from effect to cause. An expert printer examines a spoiled sheet as it leaves the press, observing that it is smeared and crumpled with a decided skew. At once he stops the machinery and puts his finger on a lever that has become crooked, or on the wheel that has been strained out of true. Mr. Joseph V. Woodworth says of milling cutters:--“When a cutter is broken by being wrongly run backwards on to the work, the breakage is characteristic. Although the man who broke it will be absolutely sure that it ran in the right direction, the cracks down the face of the teeth tell a different story.” In his manual on steel, Mr. William Metcalf reads a record equally legible to a trained eye:--“If an axe, after tempering, is found cracked near the corners of its edge, these corners have been hotter than the middle of the blade. If a crack appears at the middle of the edge, there the heat was greater than at the corners; snipping and comparing the grains will tell the story. If a somewhat straight crack is noticed, near the edge and parallel thereto, the chances are that the crack indicates a seam.”