The squaring of lunes and circular figures
Dense geometric demonstrations toward squaring lunes, with a marginal note on towers and walls.
This large sheet is a dense study in the quadrature of the circle, covering the page with dozens of lunes, half-circles, sectors and quadrants whose areas Leonardo equates through proportional reasoning. The mirror-script demonstrations argue, for instance, that a lune equals the semicircle it encloses and that similar parts of circles stand in the same proportion as their whole circles, while a marginal table lists lune areas as simple fractions (one and a quarter; a half and an eighth; a quarter and a sixteenth). Turned upside down at one edge is an unrelated note on fortification—'150 square towers of 200 feet each, the walls a hundred feet high'—with distances in stadia and the name 'Suisset.' Much of the script is not fully captured in the transcription, and the catalogue additionally tags the sheet with notes on percussion and a squirting fountain.
On this page
A lune equals the semicircle it encloses
In the right column a semicircle is marked a – d c b - m: 'the lune a d is worth the semicircle it encloses.' Removing d from the lune and b c from the semicircle leaves the triangle m equal to the remaining lune a, a step toward reducing a curved figure to a rectilinear one.
Circles in quadruple proportion; a lune as an eighth of a circle
For two circles one quadruple the other, 'the quarter of the larger is worth the whole of the smaller.' The lune a c is reckoned an eighth of a circle, and its sector likewise, so that crescent areas can be expressed as simple parts of whole circles.
Marginal table of lune areas as fractions
A numbered list gives the areas of successive figures as fractions: 'First - one and a quarter; Second - a half and an eighth; Third - a quarter and a sixteenth.' The tabulation reduces each lune or sector to a comparison of simple fractional parts.
Note on square towers and city walls
Turned upside down at the edge of the sheet is a note unrelated to the geometry: '150 square towers of 200 feet each, the walls a hundred feet high,' with lengths given as 150 and 90 stadia. It reads as a memorandum on the scale of a fortified enclosure rather than a design.
