Surfaces of solids and the squaring of the lune
Cylinder, sphere and semicircle surface ratios, quadrature of a lune, and a memorandum to the vicar
Folio 77v sets out rules comparing curved surfaces: the lateral surface of the greatest cylinder from a cube equals the surface of the greatest sphere; the whole cylinder surface is worth six of the cube's greatest circles; a sphere's surface is four times its greatest circle; and a semicircle is worth twice the greatest circle within it. Folio 74r turns to the quadrature of the lune (falcata), dividing a composite figure a c b d into portions and proving that a curved lune can be reduced to an equal rectilinear triangle, which Leonardo calls a difficult thing. Interleaved at the head of f. 74r is an unrelated personal memorandum about giving notice four days before a term expires. A faint profile head and small lens shapes appear on f. 74r but are not transcribed.
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Surfaces of the greatest cylinder and sphere from a cube
The lateral surface of the greatest cylinder that can be drawn from a cube equals the surface of the greatest sphere drawn from the same cube or cylinder. The whole surface of that greatest cylinder is worth six of the greatest circles that can be drawn from the cube's surface.
Sphere's surface equals four great circles
The surface of the sphere is worth four times the greatest circle that can be drawn from that sphere, a companion rule to the semicircle being worth twice the greatest circle it contains.
The semicircle worth twice its greatest circle
Beside a figure labelled b d a c e, Leonardo notes that the semicircle is worth twice the greatest circle that fits within it.
Dividing the greatest portion into equal quantities
The greatest portion is split into two equal quantities c L r and i t v o, with the smallest lune r and smallest portion o equal to one another. Removing the portion e and the smallest portions leaves the lune L r equal to the rectilinear triangle v, proved because two things equal to a third are equal to each other.
Quadrature of the lune a b
The surface a c b d is composed of three surfaces: b d and d c are equal in quantity but not in shape, while the lune a b is equal to neither. Subtracting d c, then again an equal value b d, leaves the rectilinear triangle a c equal to the lune a b, which Leonardo calls a difficult thing.
Memorandum on giving notice before a term expires
Opening the block on f. 74r is a note of daily business: Leonardo intends always to have given notice four days before the expiry of the term to the vicar of the provision. It is unrelated to the surrounding geometry.