Squaring lunes and semicircles, with vault studies
Geometric proofs on circle portions beside rows of dome and vault sketches
The geometric notes pursue the squaring of curved figures: a quadrant of a circle is set within a rectangle; portions and lunes labelled d c and a b are declared equal; the square inscribed in a semicircle is stated to be half the square drawn outside it; and a Euclidean-style argument (if equal things are taken from equal things...) reduces the excess of a smaller circle to a squarable triangle f. Filling most of the sheet, dozens of small sketches explore half-dome, cross- and ribbed-vault designs arranged in rows and columns. Further cursive writing on the sheet — not included here — remains untranscribed.
On this page
Quadrant of a circle within a rectangle
Along the left margin a quadrant of a circle is enclosed in a rectangle, with the direction to make one part be worth its field. It opens the sheet's series of figure-transformation exercises.
Portions and lunes declared equal (d c, a b)
Figures of circle portions and lunes (falcate) are marked d c and a b, with the note that a b c d are equal. The equality of these curved pieces is the working assumption for reducing them to squares.
The square in a semicircle is half the outer square
Beside a semicircle figure Leonardo states that the square made within the semicircle is half (subduple) the square made outside it. It is one of the proportional results underlying the squaring arguments.
Squaring the semicircle (e, b a, d c, f)
A lettered semicircle (e, b a, d c, f) supports a step-by-step proof: taking equal things from equal things, the excess of the smaller circle is matched against portions a b, and c d is removed from the smaller semicircle, leaving an excess e equal to the squarable triangle f.
Rows of half-dome and cross-vault studies
The right two-thirds of the sheet is covered with dozens of small drawings of half-domes and vaults — cross-, ribbed- and web-vault schemes — set out in tight rows and columns as a survey of ceiling and cupola designs.