Balance Arms, Friction and Counterweights
Computing the friction on a loaded balance as one-quarter of the weight, and the counterweight it needs
A page of close reasoning on how friction ("confregazione") acts on a weighted balance and how to size the counterweight that offsets it. Leonardo treats the friction as one-quarter of the rubbed weight and divides each balance arm into four parts, working the argument through fractions such as 4/3, 1/3 and 12/3. Small marginal diagrams show balances — a fulcrum drawn as a wheel with a beam and a hanging weight — beside columns of numbers, and further faint calculations run down the right margin.
On this page
Friction taken as one-quarter of the rubbed weight
Starting from the value 3 of point a, Leonardo restores the unit he had removed to reach 4/3, which he calls the true counterweight, and assigns it a quarter of friction, 1/3. He argues that the arm of friction is divided by 4 because the friction is 1/4 of the rubbed weight, so a counterweight equal to 4/3 of friction meets the need.
Dividing the two balance arms by four
Because 1/4 of the rubbed weight is placed at the tip of the friction arm, the opposite arm of the counterweight must also be divided by 4, giving four divisions on each side. He then follows the case of 7 against 4 — seven divisions where before there had been eight.
Marginal columns of numbers and fractions
The right margin carries columns of figures — 16, 48, 192, then 4, 180, 15 and repeated 192 with 12, 20, 30 and 180 — tracking the multiplications and quarter-divisions used in the balance argument. They accompany the balance sketches in the same margin.