Widening the Base of a Truncated Pyramid
Solid geometry: dressing a foreshortened pyramid in equivalent tables and cylinders
This sheet works through the solid geometry of a truncated ('foreshortened') pyramid, asking how much its base widens when it is cut down to a given height. Leonardo dresses the pyramid e f g h o in an equivalent cylinder and table, then squares each solid step by step, citing propositions 'of the first' book as he goes. The upper register carries box-like solids drawn in perspective and, at the right, a tall pyramid inscribed with converging construction lines. A closing note restates the problem: from a pyramid make a smaller one to a given lowness and find the increase of its base.
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How much a truncated pyramid's base widens
Given the pyramid e f g h o truncated in pyramidal form at height m n, Leonardo asks how much its base spreads. He dresses it in the cylinder made from its base and height, sets that cylinder at a, and by the fifth of the first book forms its table a, whose side f g equals the thickness of the first cylinder h L.
Squaring the solid at successive stations
He carries b, the square of cylinder a, to position e, turns the non-square side c toward himself, and squares it into d. Truncating the side of d at S t to the given height m n and fixing the length f in t o, he obtains the figure r t v z and finally squares the wider table to get the sought base.
Pyramid with converging construction lines
At the top right a tall pyramid is drawn in perspective, its apex tied to the base corners by ruled construction lines, illustrating the truncation and base-projection discussed in the text.
The problem restated below
A short note at the foot repeats the task: from a pyramid make a smaller one according to a given lowness, and find the increase of its base.