In the tackle-blocks that have many pulleys, it is necessary to pull the rope from the two extremes, and this is proved by the 7th proposition, which says: of the ropes of the tackles, that one will be of the slowest and shortest motion which is most remote from the tackle of the mover. Of the gravity of the excess. The adversary says that the gravity of the excess falls free; and he proves this because the weights of the balance, separated from that excess, are equal among themselves, or equally resist the descent of one another, so that among them there is neither gravity nor lightness. Therefore the excess is a power separate from the equality of those weights, and it will likewise give gravity to the weight to which it is added, and lightness to the opposite weight. It follows that such an excess descends with the same power, being joined to any of those weights, as it would if it were free in the air. It is proved, because the one of the two equal weights increases by as much as is the weight of the excess that joins with it; which excess moves the opposite weight upward as much as it would move itself downward, and so much the less as the air resists the greater body more than the smaller.

When the rope a b moves 11 braccia, the rope c d moves one braccio.

Make experiment of it.

I say make experiment of it, because although the first rope n f moves 11 braccia, it does not raise the weight more than one braccio, by means of the 11 ropes of the tackles, etc. Of the resistance of air and of balances. That body weighs less which spreads out more, and that air resists more which likewise spreads out more, that is, which with a greater sum of itself interposes between the weight and the earth. But the proportion of the weight is not according to the proportion of the air, because one spherical heavy body will have a diameter of one foot in length, and another similar spherical heavy body will have a diameter of two feet; and in this case the air interposed between the heavy body and the earth resists the heavy body of one-foot diameter by one degree, and resists the heavy body of two-foot diameter by four degrees. Therefore, from one diameter to the other it happens that the resistance of the air is quadruple; but from one weight to the other it is octuple proportion. And for this reason, the proportion of the resistances being fourfold, as from 1 to 4, and the proportion of the weight being eightfold, as from 1 to 8, the resistance of the 8 is double the resistance of the 4; and therefore the motion of the greater weight is of double velocity to the motion of the smaller, the air being of uniform resistance. But because the air doubly condenses against the mobile of double velocity, the said motions will be of equal motion. And if the motions are equal, the air will not make a double condensation, but it will be equal.