Such proportion has one density, or rarity, to the other as is the passage that one and the same arrow makes, at the same distance and force, more in the one than in the other. Measure, therefore, the motion that an arrow makes through the air, and then shoot that same one through water. But first test it through different airs, that is, foggy, rainy and clear, of the mountain and of the valley. Then shoot the same arrow into different muds, earths, stones and woods, and thus diverse materials that are not sinewy [fibrous]. And measure the passages made by one and the same arrow and by one and the same crossbow that is not tired. And such as is the length of the passage, such will be the density of the material passed through. Then make trial and comparison of the sinewy things by themselves, such as woods, cloths, wools, cottons, sponges, linen cloths, flax and the like, and then proportion them.

Among all the primary motions created by equal powers, that one will drive the moved thing farthest from itself which is made by the longest way.

The middle of the cord of the crossbow that is drawn to the place where it is nocked: such as are the degrees of its motion, such will be the degrees of the diminution of its angles.

So much does the power of the mover increase, he who draws the cord of the crossbow, as the angle that is created at the middle of that cord diminishes.

How, if the proportions are equal, the motions will be equal.

Such will be the motion made by the thing shot from the crossbow as was the angle of the cord that pushed it, and such was the motion that created that angle, and, likewise, the power of the force or of the weight that drew that cord to the creation of such angle. All the said things being of equal proportion, it is still necessary that the weight of the thing pushed by the cord, and likewise the air where it moves, be equal in its proportion, as were the aforenamed other proportions.

That is: motion of the thing pushed, motion of the cord that is loaded, thickness of the angles made by that cord when it is loaded, weight or force that pulls the middle of such cord, weight of the thing pushed, quality of air through which the pushed thing must pass.

To know how far a machine should shoot its weight.

When you make your machines, or when they are made, try different weights and different shapes of the thing that is to be shot by that machine. That is, as if such a machine were a crossbow, try with one and the same weight of arrow different thicknesses of cords and of the arms that the crossbow has. For the thinner it is, the less quantity of air resists it, and the swifter it becomes in its motions. Also try different motions between the cord and the key [trigger], because the more it is loaded, as you know, the more it drives its arrow, since it has more force. Then experiment with different weights of arrows and different weights of wood and of iron, different qualities of wood for such arrows, different thicknesses of the wood and of the iron, different lengths of the iron and of the wood. And likewise various lengths or widths of the fletchings, making them double or single or trebled, or straight or twisted. And when you have found what quality of arrow moves the greatest quantity of way, make of such experiment a general rule, and then increase proportionally when you increase such crossbow, and you will have true and good knowledge of that crossbow.