River flow, erosion, and the theory of the balance
Continuity of a river's current and bed erosion; the circular vs common balance and six definitions of statics
A densely written leaf that joins two of Leonardo's sciences. The upper half studies flowing water: invoking a continuity principle ('the eighth of the first'), he argues that where a river of even width runs faster it is shallower and where slower it is deeper, that steeper and shorter courses run faster and erode the bed more, and that surplus water overflows its banks; a stepped channel serves to prove that running water presses less on its bed than still water. The lower and left portions turn to statics, contrasting a 'circular balance' — which does not return to equilibrium by itself — with the common balance, and setting out six definitions of the arms, hangers and angles of a balance carrying equal or unequal weights. A stray memorandum about setting aside fixed goods as a provision is written among the water notes.
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Faster water is shallower: a continuity principle
In a river of equal width throughout, where the water runs faster it is shallower, and where it moves less it is deeper. This follows from a continuity argument that every part of the river's length passes an equal quantity of water in equal time, whatever its width, slope, depth or winding.
Steeper, shorter courses erode more and overflow
Where the channel is more oblique the current is faster, and faster water wears and deepens its bed while narrowing the same quantity of water. The shorter a river's course the greater its speed, and the longer the slower; where the channel's depth cannot receive the surplus, the water is forced to overflow its banks.
Running water presses less on its bed than still water
Over stepped ground, running water does not wear its bed as much as water that comes to rest after such a course. Because an element has no weight within itself, Leonardo reasons that the water wears its bed only insofar as it enters the sphere of the air, with the falling course lifting and leaping upward before falling back at d.
The circular balance does not right itself
Being of uniform gravity along any line about its pole, the circular balance does not fully perform the office of the common balance: once moved from the position of equality the common balance returns of itself, but the circular one, with weights equally heavy and distant from its centre, does not — unless, Leonardo believes, the attached weights strongly exceed the weight of the wheel.
Six definitions of the balance
The left column opens '6 definitions' and defines the arms of a balance as those ending at the hangers of their weights, and the point where weights attach. The angles the hangers make with the arms must be equal when the weights resist one another equally, and become more varied as the weights grow unequal.
Weights refer to the balance arms, not the pulley poles
For a balance beam suspended from two pulleys (labelled 2—2), the two weights are to be referred to the arms of the balance and not to the poles of the pulleys.