Triangular balance of three equal arms
Why a three-armed balance rests in any position, and how to find its double proportion
Two diagrams of a triangular (three-armed) balance study the equilibrium of a body with weights hung on branching arms. Leonardo argues that a balance of three equal arms stays at rest in whatever position it is turned, its weights always keeping a double proportion except when one arm falls on a perpendicular line, when the proportion becomes one of equality. He then gives a method: take the perpendicular through the centre of weight of each arm, compare their positions to the centre of the balance, and pair the midpoint of two arms on one side against the midpoint of the opposite arm to reveal the double proportion of spaces and weights.
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A three-armed balance rests in any position
The balance of three equal arms remains at rest of itself however you turn it, and the weights always keep a double proportion. The single exception is when one arm lies on a perpendicular line, where the proportion becomes one of equality.
Finding the double proportion of spaces and weights
Take the perpendicular through the centre of weight of each arm of the triangular balance and see how they stand relative to the centre of the balance. If two fall on one side, set their midpoint against the midpoint of the opposite arm, and the double proportion of spaces and weights appears.
Upper triangular balance (s, t, u)
The upper diagram shows a symmetrical three-armed balance with a vertical central arm and two branching lower arms, its baseline lettered s, t, u. It sets up the same equilibrium argument developed in the note below.