On the speed of ropes in pulley tackle
Studies of velocity, load and friction in multi-rope tackle, with a balance
This densely written and drawn leaf works out the mechanics of tackle blocks (taglie). Leonardo argues that if one braccio is drawn from the first of four ropes, the last rope shifts only 1/4, so the first rope runs four times faster than the last, and he carries the arithmetic through by dividing seven remaining braccia among four ropes to get one braccio and three-quarters each. He compares the relative speeds of the ropes, corrects an earlier proposition as 'false' by arguing that ropes descending from the pulleys act as the mover and bear more load than those rising, and adds fragmentary notes on friction at wheel-axles of unequal thickness. The margins carry pulley and tackle diagrams lettered a b c d e, b a, d c and f, together with a small balance.
On this page
Speed of the rope in tackle blocks
Of four ropes in a tackle, drawing the first rope by one braccio takes 1/4 from the last, while the next-to-last moves 2/4, the one before that 3/4, and the first 4/4. As these motions occur in equal time, the first rope is four times faster than the last.
Dividing seven braccia among four ropes
Ropes a b c d e are taken as four ropes of eight braccia; removing one braccio leaves seven. Dividing seven by four yields one braccio and 3/4 per rope, so that a quarter of a braccio is taken from each of the four ropes to make up the one braccio removed.
Ropes that descend bear more weight than those that rise
At the pulley b a - d c - f the rope a c holds as much of weight f as rope b d does, yet raising f by rope b d makes b d feel more weight. Leonardo marks the claim 'false' and argues by the ninth proposition that the mover always exceeds the moved: ropes descending from the pulleys are the mover, those rising are the moved.
Friction at wheel-axles of unequal thickness
Along the torn margin Leonardo notes that on the poles of equal wheels the friction is more laborious to the mover the thicker the pole, and less so for slender poles. For poles of equal thickness, a pulley of lesser width is harder to move.
Tackle, pulleys and a balance
The sheet is illustrated with tackle blocks and pulley wheels reeved with multiple ropes and with a small balance schema marked with the figures 2 and 11. These drawings carry the letters used in the accompanying analysis.