Surfaces of sphere and cylinder, and the quadrature of lunes
Comparing curved surfaces of solids and reducing crescent-shaped lunes to equal rectilinear triangles.
The verso leaf (77v) states surface relations among solids: the lateral surface of the greatest cylinder drawn from a cube equals the surface of its greatest sphere; the whole cylinder surface equals six of the cube's great circles; the sphere's surface is four times its great circle; and a semicircle is worth twice the great circle it contains. The recto leaf (74r) turns to the quadrature of lunes (falcate), dividing a figure into portions and lunes to prove that a crescent can be made equal to a rectilinear triangle — 'a difficult thing'. Amid the geometry sits a personal reminder to notify the vicar of the provision four days before its expiry. Crescents and a segmented semicircle are drawn to accompany the proof.
On this page
Cylinder and sphere surfaces from a cube
The lateral surface of the greatest cylinder that can be drawn from a cube equals the surface of the greatest sphere of that cube or cylinder; the whole cylinder surface equals six of the cube's great circles.
Sphere surface is four great circles; semicircle is two
The surface of the sphere equals four times the greatest circle that can be drawn from it, while a semicircle is worth twice the greatest circle that fits within it. These ratios frame the surface studies on the leaf.
Squaring the lune a b into a rectilinear triangle
The surface a c b d is made of three parts: two equal in quantity but not shape (b d, d c) and the lune a b. Removing d c leaves the lune, and removing an equal value again leaves the right-sided triangle a c, so the rectilinear triangle a c is proved equal to the lune a b — 'a difficult thing'.
Reminder to notify the vicar of the provision
Set among the geometry is a memorandum: he intends always to have given notice four days before the time of expiry to the vicar of the provision. It is a practical reminder unconnected to the surrounding proofs.
Crescents and a segmented semicircle
The recto is drawn with a triangulated crescent at the top and, lower down, a semicircle divided by chords into hatched segments and lunes — the very portions named in the quadrature proof. Small square-and-lune insets repeat the comparison.