Squaring the sphere's surface and branch ramifications
Solids of revolution, a plane surface equal to a sphere, and the kinds of branching
Continuing the study of solids of revolution, Leonardo describes the 'semi-composite motion' of a revolving cone or cylinder, which sweeps out a sector, a semicircle or a quadrilateral according to its proportions. He argues that an eighth of a sphere unrolls into a rectilinear triangle, so that eight such triangles make a plane surface equal to the sphere's surface, and gives (then marks 'false') a rolling method to make a circle equal to the sphere. Hemispheres, spherical caps and triangles lettered a, b, c illustrate the proofs, while a small branch diagram at the left notes three kinds of ramification. Part of the sheet lies beneath a pupil's writing.
On this page
An eighth of a sphere unrolled into a triangle
The contact of the curved side e f g h i becomes the straight line n o upon the place of equality, and the other two sides do likewise, yielding a rectilinear triangle of the same capacity as the spherical triangle. Eight such triangles make a plane surface equal to the surface of the sphere; the sides are shown rectilinear because S c from the centre S is much shorter than S a or S e.
Semi-composite motion of a revolving cone and cylinder
The semi-composite motion of a cone revolving upon a flat plane describes a sector of a circle, never a whole circle. If the cone's slant side equals its base diameter the revolution describes a semicircle, and if a cylinder's axis and diameter are equal its revolution describes a quadrilateral double the circle.
A circle equal to a sphere by rolling
Leonardo (marking one attempt 'false') proposes to mark a point on a sphere, roll it along a straight line until the point returns to the top, and take that length; making that line the diameter of a circle yields a circle equal to the sphere, whose rolling great circle equals the sphere's greatest circle.
The three kinds of branch ramification
At the left, three small branches are drawn, one numbered 6 5 4 3 2 1, with the note that there are three kinds of ramification (branching).