Widening or narrowing a pyramid's base at constant volume
Problems 17 and 18: how a pyramid's base and height trade off when its volume is kept fixed
Two numbered problems (17 and 18) continue the study of solids that keep the same quantity of material as their shape changes. In the first, the base of a pyramid is diminished to a given width and Leonardo asks how much the pyramid must grow in length, solving it by turning the pyramid into a cylinder, then a cube, narrowing the cube, and rebuilding the pyramid at three cylinder-heights. In the second, the base is instead widened and the height must shrink, ending in a square slab whose triple height equals the original pyramid. Two small pyramid figures and their bases illustrate the reciprocal relationship.
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Problem 17: base diminished, length increased
The base of a pyramid is diminished to a given width and one asks how much it grows in length. Leonardo makes the pyramid's cylinder, converts that cylinder to a cube, narrows the cube to the required thickness, and rebuilds the pyramid at three heights of the resulting cylinder so its extended form matches the diminished base.
Problem 18: base widened, height diminished
Here the base is dilated to a given width and the loss of height is sought. He makes the cylinder equal to the pyramid, reduces it to a cube, extends the cube to the given base-width, then extends its face into a cylinder to form a square slab, three heights of which equal the height of the proposed pyramid.