Geometry: doubling of areas and solids, an octagon from a square
Triangles, pyramids and star polygons, with notes on scaling and on vision past a spinning body
The sheet is crowded with geometric figures in ink and red chalk: triangles and pyramids, an octagon inscribed in a circle, hexagrams and other star polygons, a square, and a red-chalk cup or goblet. The notes state a rule that doubling a figure's diameter makes its surface quadruple and, if cubic, its solid octuple, and that to turn a square into an octagon one must throw away part of its size. A marginal note observes that a rapidly moving opaque body does not obstruct sight of what lies behind it, as a spinning reel does not hide its pole. Numeric arrays and a faint red-chalk figure are also present.
On this page
Turning a square into an octagon
To make an octagon out of the square, one must throw away part of its size. The note pairs with the drawn octagon inscribed within a circle and the several squares on the sheet.
How surface and solid grow when the diameter is doubled
The solid is more than the plane: every surface of equal figure and of doubled diameter is quadruple, one to the other, and if it is a cube it will be octuple. The rule states the square and cube scaling of area and volume with linear size.
A fast-moving opaque body does not hide what is behind it
The rapid motion of any opaque body will not in any part obstruct the things hidden behind it, like the reel (arcolaio) which does not hide its pole when it turns. The note treats how motion affects what the eye can see through the sweep of a moving object.
Triangles, an inscribed octagon and star polygons
Across the sheet are drawn triangles and pyramids of several sizes, a square, an octagon set within a circle, and hexagrams and other star polygons, together with a red-chalk cup. These figures are not individually captioned in the transcription and are described from the drawing.