Balances, Levers and the Oval Equal to a Circle
Two fragments: weighted balances and a geometric study making an oval equal in area to a circle
The sheet is made of two mounted fragments. The upper fragment (a) is filled with balances and bars hung from pulleys, each loaded with small numerical weights, together with a lever analysis that traces a 'central line' from the centre of the world through the centre of a weight and works out that 6 pounds at h will sustain 9 pounds of power at c. The lower fragment (b) turns to geometry: a large semicircle-and-oval construction for making an oval figure equal in area to a circle, a note that an oval loses its proportion as it is scaled, and an observation on how the near front of a sphere must be drawn more curved when the sphere is set in perspective. A marginal note names Theophrastus.
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Balances loaded with paired weights
A column of small balances and bars suspended from pulleys is annotated with weights such as 2 4 2, 4 4, 8-4 and 3 6. Leonardo compares how loads distributed on the arms hold or move the beam.
Central line and the lever bearing 9 and 6 pounds
Because b c is one third of the line f b, the 9-pound weight applied at site d returns into a power of 3. Since a b lifts as the half of the line of motion f b, the opposing weight must be twice as great, so 6 pounds at h will sustain 9 pounds of power at c.
Making an oval equal in area to a circle
To give an oval figure equal to a circle, the space a c of the oval must have a 'multiplication' that yields a square double to that of the space a c of the circle. The construction takes the space a b and sets it foreshortened, continuing in like manner up to p q across the lettered semicircle and oval.
An oval departs from its proportion when scaled
Removing parallel circles from the oval, or enlarging it by similar circuits, never leaves it in its first proportion. An oval 2 long and 1 high shrinks to a single line, or grown by one braccio becomes 3 by 2, passing from double to sesquialteral (3:2) proportion.
Drawing a sphere in perspective
Whoever draws a sphere in perspective errs in making the front c of like curvature to the front b, since with the eye at a the nearer front c appears less curved. To remedy this, c must be given greater curvature than the further front, in proportion to how much nearer c is to the eye.