Balancing rods by sub-double and sub-quadruple ratios
Lever rules: opposed weights equalised on paired and single beams
Four beam diagrams demonstrate Leonardo's rules of equilibrium under suspended weights. On a pair of connected rods, because segment a n is a sub-double of n m he sets 1 at m against 2 at a, and because a n is a sub-quadruple of n f he sets 4 at a against 1 at f, so 6 at a balances 1 at m and 1 at f. Further figures show a rod a b c d hung at its middle standing in balance 'by the first conclusion', and a small rod b c hung at its ends dividing its weight between the two cords. The page is headed 'Sperimentata' (tested).
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Opposed sub-double and sub-quadruple weights on rods a n m f
Because the space a n is a sub-double of n m, one is placed at m against 2 at a, making an opposed sub-double that stands in equilibrium. Because a n is a sub-quadruple of n f, 4 is given to a and one to f, so the whole is equalised and 6 at a stands against one at m and one at f.
Rod a b c d suspended at its midpoint
a b, being suspended in the middle, will stand in equilibrium by itself, 'by the first conclusion'.
Rod b c hung at its ends splits its weight between two cords
b c, being suspended at its extremities, divides its weight to each cord 'by the second conclusion', while a b rests entirely on its own cord.