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Codex Forster II — 082v

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© Victoria and Albert Museum, London

Quadruplice carrucola, con:

64 32 16 8 4

Qui si dimostra come le 4 libbre vanno sempre raddoppiando. In ogni rota che si cresce, si raddoppia il peso passato.

Colonna principale.

Se voi sapere che peso ha la corda che sostiene l'ultima carrucola, multiplica sempre cubicamente il peso appiccato dappiè col numero delle carrucole, e quel che di tal multiplicazione resulta, fia il numero delle libbre che tale ultima corda riceve di detto peso attaccato dappiè.

Diciamo adunque che esso peso attaccato dappiè sia 4, onde tu dirai: 4 libbre vie 4 carrucole fa 16 numeri. E poi di': 4 vie 16 fa 64; ed è multiplicato cubicamente, e essa corda sostiene 64 libbre per le 4 appiccate dappiè. E se esse carrucole fussino 6, diresti: 4 via 6, 24, e 4 vie 24, 981; e tanto peso sostene2 l'ultima corda delle 4 libbre attaccate dappiè.

Quadruple pulley, with:

64 32 16 8 4

Here it is shown how the 4 pounds keep doubling. With every wheel that is added, the preceding weight is doubled.

Main column.

If you wish to know what weight the cord that holds the last pulley bears, always multiply cubically the weight hung at the foot by the number of pulleys, and what results from that multiplication will be the number of pounds that this last cord receives from the said weight attached at the foot.

Let us say, then, that this weight attached at the foot is 4, so you will say: 4 pounds times 4 pulleys makes 16. And then say: 4 times 16 makes 64; and it is multiplied cubically, and this cord sustains 64 pounds for the 4 hung at the foot. And if the pulleys were 6, you would say: 4 times 6 is 24, and 4 times 24 is 96; and this much weight the last cord sustains from the 4 pounds attached at the foot.

The Quadruple Pulley and the Cubic Rule for Cord Loads

A stack of four pulleys on a post, with weights doubling 4, 8, 16, 32, 64

Physics & Mechanics Machines & Devices Arithmetic & Proportion

A tall post carries a vertical stack of four pulleys with a weight hung at its foot; the numbers 64, 32, 16, 8, 4 run up the drawing to show the load doubling at each wheel. Leonardo explains that with every added wheel the previous weight is doubled. To find what the topmost cord bears he gives a rule of multiplying the foot-weight "cubically" by the number of pulleys, worked through for a 4-pound load on 4 pulleys (giving 64) and on 6 pulleys.

On this page

Each added pulley doubles the previous weight

The series 64, 32, 16, 8, 4 marked up the drawing shows the load doubling at every wheel. Leonardo states that with each wheel added, the previous weight is doubled, so the cord supporting the uppermost pulley carries the largest share.

Multiplying the foot-weight cubically by the pulleys

To learn the load on the last cord, multiply the weight hung at the foot cubically by the number of pulleys. For 4 pounds on 4 pulleys: 4 times 4 is 16, then 4 times 16 is 64, so the cord sustains 64 pounds. For 6 pulleys: 4 times 6 is 24, then 4 times 24 is 96.

Reader's Edition — described with AI from the e-Leo transcription.