Squaring lunes: rectangles, sectors and semicircles
Area-equivalence proofs among lunes, sectors and semicircles, with animal sketches in the margin.
The sheet continues Leonardo's geometry of squaring curved figures, pairing rectangles and parallelograms with parts of circles and showing that lunes, sectors and portions of semicircles can be exchanged so that one region 'remains equal to' another. The lettered proofs turn on squarability: 'because a b c d is squarable, it follows that e f g h too is squarable.' In the left margin, unrelated to the geometry, are quick pen sketches of animals—a rearing beast and a maned, fantastical creature—beside a large brown stain. The Ambrosiana catalogue additionally tags the sheet with small firearms (scoppietti) and a note on the Mediterranean, likely among script not transcribed here.
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Squaring a parallelogram against part of a circle
The parallelogram a v c is 'set at equal value' to n b; Leonardo removes the parallelogram a to b and is left with v c, which he shows to be squarable, leaving a square n equal to it. The construction converts a curved region into an equivalent straight-sided square.
Squarability carried between two divided semicircles
For a semicircle divided as c b - d a - h g f e, 'a b c d is worth e f g h, and because a b c d is squarable, it follows that e f g h too is squarable.' Leonardo then extracts a squarable part from a known square so that its remainder equals f g.
Animal sketches in the left margin
In the left margin, unrelated to the geometry, are rapid pen sketches of animals—a rearing quadruped and a maned, fantastical creature—beside a large brown stain. These figures form no part of the transcribed geometrical text.