Squaring the surface of a sphere, and cubing the solid
One eighth of a sphere's surface generated by revolving a quarter circle
This page tackles the geometry of the sphere. At upper right two quarter-circle diagrams show one eighth of the sphere's surface generated by rotating a quarter circle, the curved parallels n o m p L q exceeding the arc by the portion labelled a c S. Below, a hemisphere (h i L K m) enclosing a cone (h i K) is used to argue the difference between curvilinear and rectilinear pyramids, to be reconciled by 'the rule of the penultimate of Pythagoras.' A sphere crossed by two orthogonal circles at lower right accompanies the closing claim that, once the surface is squared and divided into six squares, one may compose the cube and so cube the solid sphere.
On this page
Squaring one eighth of the sphere's surface
The figure a b e a c is one eighth of a sphere's surface, generated by revolving the quarter circle of the sphere's great circle. Dividing the radii into parts yields curvilinear parallels n o m p L q that exceed the quarter circle by the portion a c S. The two squarings are then joined into a single square by the rule of the penultimate proposition of Pythagoras.
Cubing the solid sphere
Once the sphere's surface has been squared, Leonardo directs the reader to divide it into six squares and assemble the cube, thereby having cubed the solid sphere. The instruction accompanies the diagram of two orthogonal circles set within a sphere at the lower right.