Pyramids, cylinders and squaring the circle
Transforming solids, the cone as a third of its cylinder, and quadrature of the circle
A geometry sheet on the mensuration and transformation of solids. Leonardo poses problems of building pyramids on given square bases, states that a cone is a third of its cylinder, and pursues the quadrature of the circle by rolling a semicircle to unroll its circumference. A long recipe converts a cylinder step by step into an equal quadrilateral tablet and thence into a pyramid. Triangles, squares, cylinders and inscribed cones illustrate the constructions.
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Making a pyramid on a given square base
From a given 'extensible' quantity Leonardo asks to build a pyramid of a given square base and find its height, or conversely to find the base for a given height. He remarks that the given and resulting quantities are irrational in themselves. Two triangles and two squares illustrate the problem.
Triangle divided into three triangles
A triangle is partitioned into three smaller triangles, lettered g, h, b, a, f, e, d, e and n. The figure supports the constructions on the sheet. Only the labels are transcribed.
A cylinder is three times its cone
For pyramids of equal base, Leonardo writes, the proportion runs from magnitude to magnitude as from height to height, and he tests this with cylinders of doubled height (marked 3 and 6). Each cone, he concludes, is a third of its cylinder, so the pyramid built on a cylinder's base contains that cylinder three times. The claim is the volume ratio of cone to cylinder.
Squaring the circle with semicircle and half-diameter
By rolling half a circle along a flat line Leonardo obtains a straight line equal to that motion, then raises the half-diameter at a right angle to it. From the arc and sagitta of the semicircle he seeks a square equal in capacity to the whole circle. This is his mechanical approach to quadrature.
Transforming a column into a pyramid of given base
In the left column the columnar (cylindrical) body is to be transformed into a pyramid whose square base matches a given square. Leonardo asks what the length of that pyramid must be. It restates the volume problem as a construction.
Reducing a cylinder to an equal quadrilateral tablet
Leonardo lays out a full recipe (lettered b, a, d, e): square the circular base by rolling half its circumference, raise the column's length at a right angle, add a uniform thickness of half the diameter, and so obtain a quadrilateral tablet equal in quantity to the column. Reworked in three such tablet-lengths, it yields the length of the pyramid. The passage converts a cylinder step by step into a rectangular solid.