A Pyramid Fills a Slab a Third of Its Height
Transmuting a pyramid into a slab, and the right-triangle cutter a d e
This leaf states a key rule of Leonardo's stereometry: the cube, slab, or cylinder built on a pyramid's base and raised to a third of the pyramid's height holds exactly the whole volume of that pyramid. From this he tackles transmuting a pyramid into a slab of stated height and finding its width. A third construction shows three bodies of decreasing height cut by a single right triangle a d e, which he says removes the same fraction (a third, fourth, or fifth) from any rectilinear body having one square side, in any proportion.
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A pyramid equals a slab a third of its height
The cube, slab, or cylinder made from the pyramid's base and extended to a third of the pyramid's height receives within itself the whole quantity of that pyramid. This is the volume identity underlying the notebook's conversions between solids.
Transmuting a pyramid into a slab
To turn a pyramid into a slab of stated height and find its width, the pyramid is first converted into its solid by the twelfth proposition of this book, by means of the third. A small pyramid-on-slab sketch accompanies the statement at upper right.
The right-triangle cutter a d e
Three bodies a b c d, e f g and h i K of decreasing height are cut with a right triangle a d e. Though drawn in rational proportions with equidistant fronts, Leonardo stresses the same triangle can cut any like body in irrational proportions, removing one same fraction, a third, fourth, or fifth, from every rectilinear body that has one square side.