Balance Beams and the Sesquitertian Ratio of Weights
Three beam diagrams showing how the weight of the arms shifts the true pivot
The upper half of the leaf carries three horizontal balance-beam diagrams, each marked with weights at points 1, 2, 3 and 4. Leonardo corrects a misjudgement: what looks like a double (2:1) proportion of weight is really a sesquitertian (4:3) one, because the arm itself weighs 2 pounds and must be added to the hung load. Dividing the span between the beam's centre of weight and the end load into three equal parts, he locates the true pole (fulcrum) at point 3.
On this page
Correcting a double proportion to a sesquitertian one
Each arm of the balance weighs 2 pounds, so the arm's own weight (2) plus the attached weight (1) together make 3 pounds. What was judged to be a double proportion of weight is therefore sesquitertian, 3 against 4. Dividing the span between the beam's centre of weight and the end load into three equal parts places the true pivot at point 3, not where the weights appear to sit.
Spaces and weights in 4:3 proportion
In the middle beam the 3 stands against the 4, a sesquitertian proportion of both spaces and weights. The apparent 2:1 (double) reading fails because the weight of the arm has not been folded into the proportion. Only when arm-weight and hung-weight are combined does the 4:3 relation appear.