Squaring lunes and curvilinear triangles; calculations
Quadrature of 'falcate' into rectilinear figures, with columns of arithmetic.
The sheet is a dense workshop of geometrical quadrature: Leonardo converts lune-shapes ('falcate') and curvilinear triangles into equal rectilinear, squarable figures, cutting the portions ab from a triangle and restoring equal areas cd, and building up one-sixth of a circle bounded by a single curve. Turned figures labelled aeb, ecd and nmo show the reasoning, with the straight side rS laid on the side tn. Around the drawings run columns of arithmetic and a measurement of a round object given as 11 1/4 braccia around, 3 2/3 in diameter, tallied to 19 7/8.
On this page
A curvilinear triangle made equal to a rectilinear surface
Removing the two portions ab from triangle e and restoring equal areas cd makes the rectilinear surface ecd equal to the curvilinear triangle aeb, provided the area neither grows nor diminishes. Placing portion d in the seat of the lune c yields a curve equal to the curvatures of a and b. Rebuilding a triangle o and restoring portions nm gives one-sixth of a circle curved on one side only.
Seeking a squarable figure and a matching lune
One figure seeks a lune n equal to the circular portion m. In the inverted corner sketch (c b d e a f), with ae equal to ef, taking f from a leaves a remainder that, with e, equals the squarable c. These are attempts at reducing curved areas to straight-sided ones that can be squared.
Measurement of a round object
A round shape drawn over a rectangle is measured: 11 1/4 braccia around, 3 2/3 in diameter, with 4 1/8 set on top and 'sleeves' of 4, the column tallying to 19 7/8. It reads as a practical reckoning of a circular piece rather than a geometrical proof.
Columns of calculation
Long columns of figures fill the margins, mixing large numbers such as 5280, 2970, 1444 and 144 with fractional entries. They appear to be running conversions and multiplications set beside the geometric work.
