Squaring Triangles and the Volume of Pyramids
A pyramid is a third of its cylinder; any triangle is half of an enclosing rectangle
The page sets out geometrical rules for measuring solids and plane figures: to find a pyramid's volume, multiply the base side by half the face-height and again by a third of the axis, resting on the theorem that a pyramid is a third of its cylinder; and to square any triangle, enclose it in a right-angled rectangle between parallel lines and take half. Ink diagrams of prisms, pyramids and a cube accompany the text. Untranscribed red-chalk sketches also appear on the sheet, including a beam-and-support mechanism at the centre-right and a further study at the lower left.
On this page
Volume of a pyramid from its cylinder
To find the solid content of a pyramid, multiply the base side by half its cathetus, then multiply the result by a third of the pyramid's axis. The rule follows because a pyramid is one third of the cylinder from which it is drawn, so taking a third of that cylinder gives the pyramid's content.
Squaring any triangle within a rectangle
To square any triangle geometrically, place it between parallel lines and construct a right-angled rectangle between them whose side is the triangle's base. Half of that rectangle is the quadrature (area) of the triangle.
Squaring an irregular pyramid
To square any irregular pyramid, set it between equidistant parallel lines with the lower line touching its whole base, then take a third of the cylinder built upon that base.
Untranscribed mechanical sketch in red chalk
A red-chalk drawing at the centre-right of the sheet shows a beam-and-support mechanism whose purpose is not explained by the transcribed geometrical text. It is included here as visible but untranscribed content, not read from the image.