Falling-body experiments and a bastion of living willows
A stream of grains resolved into a right triangle; a rampart grown from woven willow trees
The left leaf records experiments in falling motion: a stream of grains one braccio long marks descent in one unit of time, and moving the pouring vessel sideways resolves fall and transverse motion into a right triangle whose sides give the ratio of the speeds. A note begins to compound a stone's fall with a river current before breaking off. The right leaf turns to fortification, describing at length how to raise a permanent 'living' bastion by planting and interweaving willows over five years so their roots build the rampart and their branches shield the defenders. Lattice and embankment sketches accompany the willow scheme.
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Falling grains as a measure of descent in time
By experiment the line of 'discontinuous quantity' a b, a stream of falling grains one braccio long and weighing one libra, is created in a single unit of time, so the first grain descends that space in that time. Leonardo uses the falling stream to make descent measurable.
Compounding fall with transverse motion into a right triangle
Moving the pouring vessel sideways by one braccio in another unit of time builds a right triangle of two equal sides and a hypotenuse. The axis-to-base ratio a b to a c equals the ratio of the falling ball's speed to the motion of the vessel that releases it.
A stone sinking in a flowing river
Leonardo begins to compound the fall of a stone from surface to bottom with the current: if the river runs at 4 degrees of speed and the stone descends at 4 degrees, the resultant path is asked. The sentence breaks off unfinished.
A living bastion of woven willows
A permanent bastion is to be raised without timber or iron by planting willows a quarter-foot apart at m n and, year by year, banking soil and interweaving their rooting branches. By the fifth year the interlaced roots fill the rampart c d while the emptied strips b c and d e become the ditches.
Willow growth and rooting exploited for the rampart
The design relies on the willow's vigor: cut branches pressed into the soil all take root, thicken, and weave together, and the trees do not die back when their excess growth is trimmed each year. Living branches left on top screen the defenders and bar scaling ladders.
Perpendicular fall of every particle onto the hypotenuse
Once the vessel's transverse motion completes the hypotenuse of the right triangle, every particle of that hypotenuse, continuous or discontinuous, falls by a perpendicular line beneath the whole line of transverse motion. The rule ties the compounded motion back to vertical descent.