Finding weights from their spacing; bounce of a rolling ball
A rule to deduce cord loads from position, and why a rolling body rebounds farther than a flying one
When the positions of the loads are known but not their magnitudes, Leonardo gives a rule: the opposite weights taken by two cords stand in the same proportion as the spaces between the beam's centre and the cords. He works a case of five spaces divided by the beam's eight partitions (each named 8/5), assigning a b the value 4 4/5 and a the value 3 1/5. A separate note contrasts motion along the ground with motion through air, claiming that a body rolling and bouncing on the ground rebounds farther than one thrown through the air even when the driving forces are equal. Rows of weighted beams and two touching circles accompany the reasoning.
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Weights follow the proportion of their spacing
Whatever proportion the spaces have that lie between the centre of the beam and the two cords that suspend it, such is the proportion of the opposite weights those cords take of the beam, at n. This lets one recover the loads from position alone when the weights themselves are unknown.
Sharing five spaces by the eight-part rule
Between the 2 cords there are 5 spaces, divided by the beam's eight partition so each is named 8/5. Giving 3 to a b and 2 to a: the 24/5 of a b make 4 4/5 and the 16/5 of a make 3 1/5.
A rolling body bounces farther than a flying one
The bounce of a thing rolling along the ground is greater than that of a thing moving through the air, even when the original motions and driving forces are equal. This is because rolling balls do not seek the curved line of descent that a body falling through the air must trace.