Flattening the Eighth Part of a Sphere into a Square
Quadrature of spherical surfaces among many small cap and vault-like figures
A torn sheet crowded with small pen sketches of curved caps, spherical segments, vault-like domes and triangles, worked around the problem of measuring a curved surface. Leonardo proposes that parallels drawn on the eighth part of a sphere, brought into contact lengthwise and crosswise, describe a plane figure equal to that octant. He treats three equal octants a b c, d e f and g h i and, by rectifying the periphery a c into a straight line by an earlier rule, forms a square K L m n equal to a quarter of the sphere's surface. Further faint untranscribed text accompanies the diagrams.
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Flattening the eighth part of a sphere
Parallels drawn upon the eighth part of the sphere, then brought into contact lengthwise and crosswise, will describe a plane figure equal to that eighth part of the sphere. It is the governing proposition for reducing the curved octant to a flat area.
A curved-sided triangle from revolved parallels
A left-hand triangle with one curved side f h L, its sides created by the revolution of curved sides; the curved side f h arises from parallels born of that revolution. The passage is fragmentary, with several lacunae in the transcription.
An octant squared as a quarter of the sphere's surface
Octant a b c, similar to d e f and g h i, is given as much surface again to form a square like K L m n. This square equals a quarter of the sphere's surface, obtained by extending the periphery a c into a straight line by the rule already given.