The science of the balance: weights, arms and pivots
Beam scales with proportional counterweights and notes on suspension and equilibrium
Four rows of balance-beam diagrams hang weighted pans (drawn as small circles) from graduated arms, with numbers such as 1, 1 1/3, 1 1/2 and 2 recording the counterweights that hold each beam level. Leonardo reasons that a load spread along an arm can be reduced to its midpoint for reckoning equilibrium, and that any thread carrying a weight straightens to its perpendicular. A closing memorandum urges experiment on the nature of a balance's pivots, thick or thin, and on where beams should be lashed.
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Equilibrium of weights reduced to the midpoint of an arm
For every 1 at f there stands 1 1/3 opposite at a; 1 at e balances 1 1/2 at b; 1 at d stands with 2 at e. If the load a b c is drawn wholly to its middle b, it balances d e f drawn to its middle e. The nearer a support is to its fixed point, the more it carries.
A loaded thread aligns to its perpendicular
Rods hung with weights of 8, and 4 with 4, 4 with 6, 4 with 5 test the suspension: every thread attached to bear a weight straightens itself to its perpendicular.
Programme to test the pivots and lashings of balances
Leonardo sets himself to experiment and record the nature of the poles of balances, whether they be thick or thin, or set at the middle, below, above or shared; and likewise the lashings of beams, whether tied at the middle or at the side.
Numerical ratios of the counterweights
The beams are annotated with proportional weight values such as 1, 1 1/3, 1 1/2 and 2, the arithmetic of the ratios that keep each arm in balance.