Protection against the shot of a bombard.

For driving a pile obliquely.

4 measures – 3 – 2 – 1 – 1/2 – 1/4.

a 4 b e – 2
a b is the way that the moved thing is driven by its mover, and the figures of the numbers denote the time that this mover accompanies the thing it moves; and b c is the road, or the space of the motion, that the moved thing makes after it has left its mover.
Among arrows accompanied by their movers over equal distance, you will find in the motions they follow such proportion as was that of the time during which their mover accompanied them.

When ...
The motions of the arrows will have among themselves such proportion as will be that of the angles made by the cord.
And the weights of the arrows ought to have among themselves such proportion as was that of the weights that loaded them.

0 – 200 – 300 – 400 – 500
The motions will have such proportion as will be the weight that loads the crossbow.
Devise a crossbow that can be discharged at any draw, that is, so that the nut travels along the cord.

Spring.
A spring that clamps in a circle.

If the moved thing be accompanied by its mover a certain quantity of road, the proportion of the motion that follows will be as the time during which they were accompanied by their mover — that is, the proportion of the following motion will be as the velocity and time of its mover. And the velocity of the mover will be as the quantities of the weights that loaded such a machine.
Among machines of equal proportion of members, that one will cast the moved thing farthest which is loaded exactly with the greater weight. And the proportion of the motions will be as that of the powers that move and of the weights that cause such powers.

a b 400 – c d 800
The cord that makes the motion a b moves with 400 pounds and draws 400 braccia, which 400 pounds exactly finish their power against the resistance of the crossbow's cord when the cord reaches the nut. Now the motion of the other crossbow is the very same, but it is loaded with double weight.

Of the crossbow again.
First examine a crossbow that shoots an arrow proportioned to its force, testing lighter and heavier ones, of iron and shaft or of iron only or shaft only, and of various fletchings, long and short, twisted and straight, and of various materials; and likewise do with the arrow's shaft, of various lengths and thicknesses and woods and weights. Then, after you have seen which arrow goes farthest, make a rule upon it, considering well its qualities. And if that crossbow brought its cord exactly to the nut with a weight of 400 pounds, know that another crossbow of exactly the same proportion, loaded with such motion of cord as the first and with doubled weight — surely, if the arrow be proportioned to that force as the first was to the first crossbow — that second arrow will go twice as far, more heavily loaded ...

The cords of crossbows that make one and the same motion and that are loaded with greater weight: the proportion of their motions will be as that of the weights that loaded them.

Material motion is made by weight or by force. That which is made by weight sometimes generates force and sometimes weight; and that which is made by force does the like. Force is created by the increase or diminution of its cause.