Weights, plumb-lines, and squaring the circle

Definitions of the plumb-line, perpendicular, and central line

The plumb-line (appendiculo) is defined as a material straight line, or cord, that by its length holds a weight suspended in the air and joins the central line of that weight. The central line runs from the center of gravity of suspended bodies to the center of the world, and the perpendicular is the line that always falls upon another at right angles.

Balances with weights and ratios of equilibrium

Three figures show a body hung from a beam with the ratios 4/2, 4/3 and 4/4, and the site of equality is defined as the straight line above which a plumb-line falls midway between two right angles. They set out the geometry of a balance in equilibrium.

Squaring a quarter of a circular parallelogram

By removing a triangle from a sector and an equal portion from a curvilinear parallelogram, then rejoining and dividing the pieces, Leonardo forms the fourth part of a circular parallelogram equal to a known triangle, and so squarable. He claims that if its proportion to the remainder is rational, a sector, and thence a circle, equal to it can be given, a thing never done before his time.

Number tables and ratios

Small tables of figures accompany the drawings, among them 15 multiplied by 15 giving 225, and the ratios 4/2, 4/3 and 4/4 attached to the balance figures. They record the calculations behind the mechanical and geometrical proofs.

Foglio 261 a Recto

Foglio 261 b Recto

Figura a sinistra:

Figura centrale:

Figura a destra:

Figura con:

Figura di corpo sospeso:

Figura di corpo sospeso:

Due figure a forma di T:

Foglio 261 c Recto

Figura:

Figura:

Figura:

Sheet 261 a Recto

Sheet 261 b Recto

Figure on the left:

Central figure:

Figure on the right:

Figure with:

Figure of a suspended body:

Figure of a suspended body:

Two T-shaped figures:

Sheet 261 c Recto

Figure:

Figure:

Figure:

Weights, plumb-lines, and squaring the circle

On this page

Definitions of the plumb-line, perpendicular, and central line

Balances with weights and ratios of equilibrium

Squaring a quarter of a circular parallelogram

Number tables and ratios