The true centre of gravity of a balance's arms
Three numbered propositions on arms as mutual counterweights
Three numbered propositions, each with a beam diagram, treat the arms of a balance as bodies with their own weight. The first states that the midpoint of each arm's length is the true centre of its gravity; the second, that the arms act as mutual counterweights whose effect varies with the proportion of the arms. The third, illustrated by a beam labelled a m n, gives a worked case: since m n is a subtriple of m a, the arm a m is lightened by 1/9 of its weight.
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Midpoint as the true centre of gravity
The centre of the length of each arm of the balance is the true centre of its gravity. The arms, moreover, each make of themselves a counterweight to the other, and these counterweights vary with the arms as their proportions vary.
An arm lightened by 1/9 of its weight
The proportion one arm of the balance has with its opposite is the same proportion the weight it lightens on that opposite arm has with it. Because m n is a subtriple of m a, a weight subtriple to m n is lightened from m a, so that a m lacks 1/9 of its weight.