The tip of a pyramid: squaring and cubing its fractions
A quarter of the height gives 1/16 of a surface, but a millionth of a solid.
Leonardo gives a rule for how much of a pyramid its tip contains when cut at a given fraction of the height. For a surface (triangular) pyramid you square the denominator - a quarter of the height yields 1/16, a third yields 1/9 - while for a solid pyramid you cube it, so the hundredth of the height is a millionth of the whole (1). Inversely, the millionth part corresponds to the cube root of a million, 100, that is the hundredth of the height. He extends the method to pyramids with two curved sides by first solving the rectilinear case and then applying the measure to the curvilinear one, using the square and cube roots of 4 laid on the axis. Triangles subdivided into smaller triangles at the right illustrate the squaring of the quarter into sixteenths.
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Quarter of the height gives a sixteenth of a surface pyramid
To find what part of a surface pyramid its tip is, multiply the denomination of the request by itself: for a quarter, 4 times 4 makes 16, so the tip cut at a quarter of the height is 1/16 of the whole. Likewise a third of the height gives 1/9.
Solid pyramids: cube the denominator
For a solid pyramid the denomination is raised cubically, not squarely. The hundredth of the height gives 100 times 100 equals 10,000, and 100 times 10,000 equals a million, so the hundredth of the height is a millionth of the whole; inversely the millionth part answers to the cube root of a million, 100.
From rectilinear to curvilinear pyramids via roots
For pyramids with two curved sides the height of any required part is found by first solving the rectilinear pyramid and applying that measure to the curvilinear one. The quarter of a surface pyramid lies at half the height (the root of 4); for a solid, the cube root of 4 is laid on the axis toward the tip, and this root is the whole cathetus of that part.
Subdivided triangle diagrams
Along the right margin a large triangle is ruled into rows of smaller triangles, with two lesser subdivided triangles beneath it. The figures illustrate the squaring of the quarter of a triangular pyramid into a sixteenth.