Squaring the circle through sectors and lunes
Equal-area constructions of circular sectors; further prose on the sheet
A dense sheet crowded with circular sectors and sickle-shaped lunes, all serving Leonardo's pursuit of the quadrature of the circle: he notes that making one segment equal to another would yield the squaring of the whole. The lettered constructions repeatedly transform curved figures into equal-area combinations — matching sectors of equal periphery (a, e and b, c), building compound equalities (f g equal to a e and b c), and comparing lunes cut from circles in a 1:16 ratio (m n o b a c n). A recurring theorem holds that from two equal continuous surfaces, one whole and one a part of another, an equal part may always be removed, to infinity. Beyond these geometric captions the sheet carries further dense mirror-script text not transcribed here; the Ambrosiana catalogues the leaf under themes of rivers, canals, boats and their floods.
On this page
Squaring a segment, and by consequence the circle
Along the upper margin Leonardo observes that if you could make b equal to a you would obtain the squaring of a, and consequently the squaring of the circle. The constraint is that the cut cannot be made in the direction that severs a b.
Two sectors of equal periphery and equal area
Given b with a periphery matching b c, and a of smaller periphery but equal to b, he adds the surface e equal to c, thereby producing two surfaces equal to one another and of the same periphery. The figures are keyed a – e and b – c.
Combining sectors: f g equal to a e and b c
In a further sector keyed f – g, f is set equal to a and to b above, and g equal to e and to c above, so that the whole f g equals both a e and b c. The construction chains the earlier equalities into a larger equal figure.
Lunes from circles in a 1:16 ratio
Two sectors of different circles are set side by side (m n o b a c n), where the curvature of b belongs to a circle sixteen times smaller than that of a c n. By subtracting equal parts c and matching the portions d and m, Leonardo isolates a squarable part equal to the otherwise unsquarable b.
Removing equal parts from continuous surfaces
From two surfaces equal in quantity — one whole and the other part of another — an equal part may always be removed, and this can be repeated successively to infinity because they are continuous quantities. The principle underlies the sheet's repeated sector subtractions.
Further untranscribed prose on the sheet
Beyond the geometric captions transcribed here, the crowded sheet bears additional dense mirror-script text that is not included in this transcription. The Ambrosiana catalogues the leaf under themes of rivers, canals, boats set along currents, and the terrible aspects of their inundations.