Rods Bending Under Weights, and the Circle Defined

A 12-braccia rod bows one braccio under a pound

Leonardo reports finding that a rod of 12 braccia, with one pound hung at its middle, makes one braccio of curvature. He then poses the inverse: what weight would a rod of six braccia of like thickness need to create the same braccio of arc.

Strength in proportion to length

The strength of a rod, he states, holds the same proportion as its length, tabulated as 1, 8, 64, 512. Accordingly a single rod of 6 braccia is said to be twice as strong at its middle as four rods of 12 braccia of like thickness bound together.

Equal weights, equal degrees of deflection

A rod fixed at one end and describing, as it lowers, the twelfth part of a circle deflects in equal degrees when the degrees are made by equal weights. Likewise the rod a b c, until its curvature reaches the sixth part of a circle, descends in degrees equal among themselves under equal weights.

Moving a weight to keep the same curvature

Hanging a weight at the middle of a horizontal rod, Leonardo proposes shifting it to five successive sites nearer an end, each time doubling the previous weight, so that the rod keeps its first curvature. He offers, given the middle weight that produces a certain curve, to name the weight needed at any point you touch to reproduce it.

Definition of the circle

A circle is a plane figure contained by a single line, called the circumference or periphery, in the middle of which is a point or centre. The figure marks that central point.

Radii, diameter and chords

All straight lines drawn from the centre to the circle are equal; the greatest line that can be placed within it is the diameter, which cuts the circle into two equal parts. In a further figure (o, m, n, a, f), of lines drawn from a point off the centre the longest bends most toward the centre, the shortest is most remote, and those equally distant from the centre are equal.

Facciata destra. * Aste di varia lunghezza incurvate da vari pesi:

Fascio d'aste ricurve, coi pesi:

* Aste ricurve, con:

Asta e fascio di quattro aste ricurve, con:

Doppia figura di aste piegate, con :

Asta ricurva con:

Asta orizzontale e la stessa incurvata di otto gradi, con:

Asta sospesa a due corde e curvata da otto pesi, con:

Asta con quattro curvature, con:

Sotto la precedente, asta ricurva con:

Asta ricurva fissata a un estremo, con:

Quattro aste come sopra, con:

A sinistra, numeri

Facciata sinistra. Cerchio col centro segnato:

Cerchio con cinque raggi:

Figura con:

Aste con curvità concava o convessa, con:

Right-hand side. * Rods of various lengths bent by various weights:

Bundle of curved rods, with the weights:

* Curved rods, with:

A rod and a bundle of four curved rods, with:

Double figure of bent rods, with:

Curved rod, with:

A horizontal rod and the same bent by eight degrees, with:

A rod suspended by two cords and curved by eight weights, with:

A rod with four curvatures, with:

Below the preceding, a curved rod with:

A curved rod fixed at one end, with:

Four rods as above, with:

On the left, numbers

Left-hand side. Circle with the centre marked:

Circle with five radii:

Figure with:

Rods with concave or convex curvature, with: