Solid Geometry of Pyramids, Cones and Cylinders
Dividing a pyramid into eight, its volume as a third of the cylinder, and an oval lathe
This crowded sheet is filled with red-chalk and pen studies of solids — pyramids on square bases, pyramids set within cubes and cylinders, cones and prisms. Leonardo works out that every pyramid on a square base resolves into eight similar pyramids, that when cut halfway up it splits into parts in 'sevenfold proportion', and that a pyramid equals one third of its enclosing cylinder, which yields its cubic content. Lettered figures (a, b, c, d, e) accompany the reasoning. A separate line among the diagrams reminds him to 'have the German make the oval lathe.'
On this page
Every pyramid on a square base resolves into eight similar pyramids
Above the largest drawing Leonardo asserts that every pyramid built on a square base resolves into eight pyramids similar in figure to the whole. He adds that such a pyramid, cut parallel to its base at half its height, is divided into two parts standing in sevenfold proportion.
A pyramid is one third of its cylinder
Working from a lettered figure (e a d, g f, c b), Leonardo states that because the pyramid is one third of its cylinder, dividing the cylinder's height into three gives the third that equals the pyramid's cubic content. A companion note across two pyramids-within-cubes says that the one body is worth the whole other pyramid.
Proportion of pyramids by their bases and axes
Leonardo records proportional rules for solids: pyramids of equal axes stand to one another as their bases, and pyramids of equal base as their axes. He also considers a body whose height is a third of the pyramid and whose base equals the pyramid's base.
Reminder to have the German make an oval lathe
Amid the geometry, a practical memorandum in pen reads: have the German make the oval lathe. It is one of Leonardo's reminders to a craftsman, here for a lathe capable of turning oval forms.