The motion of every free descent made by heavy bodies is toward the centre of the world.

If the motions are equally swift.

The instant of central motion being indivisible, the time in which that motion continues is divisible; it is necessary that the line g f can never be set as an equidistant (level) line.

a b is equal to a c; d e f cuts in half each of the sides of the triangle; g h i cuts at the fourth of each of those sides of the triangle.

If the line of the motion made by the mover and the line of the first moving body are at right angles at their creation, the line of the positions of all the other moving bodies will be straight, as r f g shows; and this happens when the motion of the moving body equals the motion of the mover. And these triangles can occur in several ways, that is: the first is when ...

No motion is of such great slowness that in it a point of rest may be found.

Therefore, the motion being divisible with time, the transverse motion can be divided; made in more than one instant, the motion made in more than one point is divisible.

One motion has to another the same proportion as the level line has to the perpendicular.

However slow a motion one may imagine, there will not be found in it any part of time so small that it can come to rest, that is, in a single instant.

It will be caught at the seesaw.

Water that falls neither thins nor diminishes its quantity in any part of the length of its descent.

Therefore one may say that in every part of the length of the motion of water moved by itself, an equal weight of water passes in an equal time.

For the adversary.

This is proved true by showing that, in that part where you would wish to say that more water passes, or that more water enters than that which leaves, you would have to confess that in such a place all the element of water would in time be gathered; because at every point where two casks of water entered and only one cask were poured out, you would accumulate in that place as many casks of water as the points you could count in the time.

The adversary.

The adversary says that in the courses of waters there are many places where, by its swift course, the water thins in such a way that ...

You have to say whether the slowness is before or after the swiftness, granting that the slow is slower than the original and the swift again is swifter than the original, or that the original is midway between the swiftness and the slowness of the two aforesaid actions.

Or, to return to the subject: if you say the swift is before the slow, it will bring so much water to where it slows that in time it will touch the sky. And if you say the slow is before the swift, the swift will consume the slow and the original that was between the motion of the slow and the swift. And if you say the original is swifter than the two aforesaid, these two will receive more water than they can clear away, whence they will greatly rise or dilate.

The proportion which the side a b has to the side a g, the same has the transverse motion to the natural motion.

When the mover of the sand, which goes sowing it with equal outpouring along its whole path, is swifter than the descent of that sand, then in an equal time the motion of that mover will be longer than the descent of the sand by as much as it is swifter.