Turning a Cube into a Taller Pyramid of Equal Volume
Problem 10: how much narrower the pyramid's base becomes than the cube's
Leonardo poses a problem of solid geometry: from a cube, make a pyramid that rises to a given height greater than the three sides of the cube laid end to end, and determine how much narrower the pyramid's base must be than the cube's base. The construction treats it as a volume-preserving transformation, building the pyramid's 'natural cylinder' (the solid on its base reaching a third of its height) and then trimming it with a right triangle. The upper half of the sheet is filled with tall narrow pyramids drawn over a common base line, with a boxed cube at the left, matching the caption's 'five pyramids variously arranged.'
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From a cube to a taller pyramid of equal volume
A cube is to be turned into a pyramid rising to a given height greater than the three sides of the cube joined end to end. The question asked is how much narrower the pyramid's base becomes than the base of its cube. Leonardo frames this as an exact conversion of one solid into another of the same content.
Right-triangle construction that trims the base
Starting from the cube a b e f, he builds the pyramid's natural cylinder L m f g, which reaches a third of the pyramid's height and exceeds the cube b c f g. He then draws the right triangle L f h and its hypotenuse L h to cut the cube's corner at c; the line of excess b h divides the pyramid's base at i, and the surplus is cut away.
Five pyramids drawn across the sheet
The upper part of the page carries several tall, narrow pyramids set over a shared base line, together with a boxed cube at the left carrying letters. The arrangement illustrates the caption's phrase 'five pyramids variously arranged.'