Squaring a circular sector; balance and pulley mechanics
Quadrature of a sector, the centre of gravity of suspended weights, and cord thickness in pulleys
The page opens under the heading "Geometry" with a quadrature demonstration: to square an unsquarable surface b, Leonardo borrows a circular sector c that is itself squarable by its motion, joins the two, then subtracts a square equal to the lent sector. Below, a series of balance diagrams and a long "principal column" argue that the centre of gravity of two suspended weights b and d lies on the central line of the supporting cord, midway between the hangers, using a sesquialteral proportion of arms and weights. A final note on a pulley explains that a thicker cord moves its wheel more easily because the lever of friction is longer from the centre of the thick cord. Further geometric and mechanical figures accompany the text.
On this page
Squaring a surface by lending a squarable sector
To square the surface a it is necessary to lend the sector c, which by its own motion is squarable in itself. Adding the squarable c to the unsquarable b makes c b a squarable surface, from which one separates a square equal to the lent sector c, leaving the remainder squared.
Centre of gravity of two suspended weights
The centre of gravity of a suspended body lies on the central line of the cord that supports it. Proved by weights b and d hung from the balance: their combined centre lies in the middle between the two hangers, since weight a resists b and weight c resists d on equal arms, the spaces n m and m p standing in sesquialteral proportion to the inversely proportioned weights.
A thicker cord eases the pulley's motion
The cord of greater thickness makes its pulley move more easily. This happens because the lever of the friction is longer from the centre of the thick cord to the side of the friction than from the centre of the thin cord.