Windlass and cord mechanics with quadrature of lunes

Cord thickness, distance and the labour of the windlass

Leonardo states that the part of a cord's thickness farthest from the centre of the windlass gives the most labour to its mover, while the centre of the cord's length is unchanged by twisting. If the beam is double the thickest cord and quadruple the thinnest, the load on the thicker cord rises a sixth faster; if the greater cord equals the beam and is double the smaller, its load rises a fifth of the time sooner while the mover's effort grows by 1/10.

Cords wound on the windlass-beam (subbio)

A figure of cords wound on a windlass-beam carries the rule that the strands of a cord supporting a weight are the more loaded the nearer they lie to the centre of gravity of that weight.

Parallel spaces proved equal in a lettered figure

In a pyramid lettered a L e b, n g f d o c, h K i, the two parallel spaces c i, d K and g h are proved equal because c d equals d g and i K equals K h, with equal corresponding angles and equal diameters. In squaring the whole to a i K h, what the perpendicular a K cuts from d e f is regained on the opposite side in g n h.

Squaring a sixth-of-a-circle sector

In a figure lettered c b a, a b c is set equal to b d, with a b equal to b c and equal to one sixth of a circle (a sector). The part a is squarable, and the unsquarable part c is made known by being equal to a; the curve of c is an eighth of a circle.

Numeric schemes for balances and cords

The right half is studded with numeric schemes attached to balances and to cords on the windlass, for example the figures 6, 6 / 81 / 12 / 6 beside one balance, recording the weights and ratios Leonardo set against one another.

(a) Metà destra. Schema di bilancia con:

Numeri e operazioni aritmetiche:

Presso la piega mediana, schema applicato a un subbio con corde:

Più a destra, altro schema:

Altro schema:

Metà sinistra. In alto:

Sopra la piega mediana, schema con:

Sotto il precedente, schema e numeri:

Più in basso:

Riprendendo dall'alto tra schemi di corde e subbi:

Sezioni di subbio e corda:

Schema di bilancia:

Altri numeri:

In basso presso la piega mediana. Figura di corde avvolte a un subbio:

Andando a sinistra:

(b) Angolo superiore destro, con riferimento alla figura successiva:

Piramide colle lettere:

Altre due figure simili, con:

Lungo il margine sinistro:

Figura simile alle precedenti, con:

Lungo il margine destro, in basso, figure colle lettere:

Andando verso sinistra, falcata colla lettera:

Figura di piramide e la metà inferiore della stessa, colle lettere:

Sotto la precedente, figura con:

Verso sinistra, figura simile alla predetta con:

Sopra la precedente, figura di semicerchi sovrapposti a un cerchio, con:

Sopra la figura predetta:

Sul margine inferiore, operazione aritmetica:

(a) Right half. Diagram of a balance with:

Numbers and arithmetical operations:

Near the central fold, a diagram applied to a windlass-beam with cords:

Further to the right, another diagram:

Another diagram:

Left half. At the top:

Above the central fold, a diagram with:

Below the previous, a diagram and numbers:

Lower down:

Resuming from the top, among diagrams of cords and windlass-beams:

Sections of windlass-beam and cord:

Diagram of a balance:

Further numbers:

At the bottom, near the central fold. Figure of cords wound on a windlass-beam:

Moving to the left:

(b) Upper right corner, referring to the following figure:

A pyramid with the letters:

Two more similar figures, with:

Along the left margin:

A figure like the preceding, with:

Along the right margin, below, figures with the letters:

Going toward the left, a lune with the letter:

A figure of a pyramid and the lower half of the same, with the letters:

Below the previous, a figure with:

Toward the left, a figure like the aforesaid, with:

Above the previous, a figure of semicircles superimposed on a circle, with:

Above the aforesaid figure:

On the lower margin, an arithmetical operation:

On this page

Cord thickness, distance and the labour of the windlass

Cords wound on the windlass-beam (subbio)

Parallel spaces proved equal in a lettered figure

Squaring a sixth-of-a-circle sector

Numeric schemes for balances and cords