Pyramids and cubes: proportion, weight and the doubling of the cube

A pyramid cut by a plane parallel to its base

A triangular section cut parallel to the base yields a smaller triangle whose sides keep the same proportion as the sides of the triangle it was divided from. Cut at the middle of the height, the length of the cut is half the length of the base. The figure is lettered a, i, k, h, g, e, d, m, f, c, b.

Square-based pyramid is double the triangular-based one

Of pyramids of equal height and equal side-length, the one on a square base holds twice the quantity of the one on a triangular base. This governs the family of solids drawn across the sheet.

Cube root of a pyramid found as for cubes

The cube root of the pyramid of four bases is found just as for cubic bodies: 2 times 2 makes 4, and 2 times 4 makes 8. In the greater pyramid b c g d f sit four equal pyramids born from doubling the square root of 4, the remainder making 16, an eighth of which is the suboctuple part.

Squaring a pyramid through its cylinder

To square this pyramid, take a third of the height of the cylinder from which the pyramid issued and you obtain its quadrature; or build a square on the pyramid's base that is high by a third of its axis. Parallelepipeds are drawn beside pyramids of equal volume to demonstrate.

Dividing a pyramid into equal weights

Of two equal pyramids a and b, Leonardo would make one the pyramid taken from the middle forward and the other from the middle backward, so that their division answers his aim of obtaining half a pyramid by weight.

Duplication of the cube by equal weights

Cube the whole, take the half and re-cube it, then make from that half a pyramid proportioned to the greater one; the remainder toward the base weighs the same as the smaller pyramid. By this the duplication of the cube is achieved, lending a half or two-thirds to a pyramid to turn it into a quadrilateral cylinder and back.

(a) A destra:

Sotto a tre figure di piramidi:

Alla sinistra delle predette figure:

Metà sinistra, dall'alto. Figura di piramide colle lettere:

Due triangoli tagliati da una parallela alla base, colle lettere:

Due piramidi colle lettere:

Sul margine inferiore, due quadrati scomposti in quadratini, e due triangoli scomposti in 16 triangoli simili, con:

(b) A destra dall'alto in basso. Triangolo o sezione di piramide con:

Figure di parallelepipedi e piramidi equivalenti:

Parallelepipedo diviso a metà colle lettere:

Parallelepipedo e piramide equivalente colle lettere:

Colonna centrale. Due parallelepipedi colle piramidi equivalenti e più in basso sezione di piramide divisa, colle lettere:

Colonna sinistra. Piramide capovolta e cubo diviso a metà da un piano orizzontale:

In basso, sezione di mezza piramide suddivisa e operazione aritmetica:

(a) On the right:

Beneath three pyramid figures:

To the left of the aforesaid figures:

Left half, from the top. Pyramid figure with letters:

Two triangles cut by a line parallel to the base, with letters:

Two pyramids with letters:

On the lower margin, two squares broken into small squares and two triangles broken into 16 similar triangles, with:

(b) On the right, top to bottom. Triangle or pyramid section with:

Parallelepipeds and equivalent pyramids:

Parallelepiped divided in half, with letters:

Parallelepiped and equivalent pyramid, with letters:

Central column. Two parallelepipeds with equivalent pyramids and, lower, a divided pyramid section, with letters:

Left column. Inverted pyramid and cube halved by a horizontal plane:

At the bottom, section of a half-pyramid subdivided, and an arithmetic operation: