Example.1

Verbi gratia.2 If there is a location arranged as three different widths, connected together, where the first3 and smallest width enters four4 times into the second and the second5 enters two times into the third, then I will say about the men who fill these locations with their persons — and they need to be walking continuously — that when the men of the largest place take one step, those in the second smallest6 room will take two, and those7 in the third place, which is one fourth the size of the second place, will take eight steps in that same time.  And you will find this proportion in all movements that pass through locations of differing widths.8

Note the squirting implement.  When its male that shoots the water out9 moves the distance of one finger, the first water that came out went a distance of two braccia.  And thus, if it were possible to lift a vessel the size of ten barrels up to the height of ten miles, you would find that, with the universal outflow continuing, the first wine that came out of the vessel would have gone the ten miles before the surface of the wine dropped10 by two braccia.  You will find this same effect in the movements of gearwheels with their pinions, since if the axle of the gearwheel is the same size as the pinion, then in the turning of this gearwheel, the movements11 of the pinion and of the circumference12 of the gearwheel will be as much faster than that of its axle,13 as the circumference of the pinion enters into the circumference of the gearwheel.14

Any movement of water of equal width and surface15 will flow as much stronger in one location than in another, as the one location is shallower than the other.

a b – c d – e f – m n

[III]

This proportion is clearly proved, since, although the river is of equal width and surface, it is not of equal depth, so it is necessary that the course of this river, also, be of unequal movement, for the reasons mentioned above.   Its movement is of this quality.  Suppose that the figure to the side with markings is the river.  I say that the water at m n is as many times swifter than that at a b, as m n enters into a b, which four times.  Therefore, the water at m n travels four times as fast as it does at a b, and three times as fast as it does at c d, and two times as fast as it does at e f.

Every body of water whose surface moves in waves16 toward a given direction will move with contrary course at its bottom.

This reason is proved in this manner, that is: We see the sea sending its waves toward the shore, and although the wave17 that terminates on the shore is the last of its company, it is always superseded18 and submerged by the next to the last.  Nonetheless, this next to the last does not pass beyond the last, indeed, it submerges at the same place as the last one.  As this submerging is, thus, always in continual motion where the sea19 borders with the shore, it is necessary for there to be a contrary motion after that, at the bottom of the sea.  And as much water returns underneath, toward the cause of its movement, as is driven at the upper part, by the water's motor.20