Weight on Two Cords; Squaring and Cubing the Sphere
Force-triangle of a suspended weight, and the geometric quadrature of curved figures
The page opens under the heading 'On weights' with a rule for a heavy body hung from two oblique cords: the angle between the cords, cut by the vertical through the body, is divided in the same proportion as the weight is shared between the two supports. The rest of the folio turns to geometry, giving constructions 'by motion' for squaring the surface of a hemisphere, cubing the sphere by filling it with equal pyramids, and squaring a circular segment. Marginal diagrams show a force-triangle, stacked pyramids, a subdivided cube, a circle broken into triangles, and a circular segment.
On this page
Weight suspended from two oblique cords
If the angle formed where the two oblique cords meet at the hanging weight is cut by the vertical line through that weight, the angle is split in the same proportion in which the weight is shared between the two cords. The same proportion holds between base and base, angle and angle, and whole triangle and whole triangle, and mirrors the loads the weight delivers to its two supports.
Squaring the surface of a hemisphere (a, e, b, c, d, n)
To square the surface of the hemisphere a e b c, draw the chord a b of the arc a e b, then the line c n parallel to a b, and transfer the angle a c b to the point d on that line, obtaining a quarter-circle a e b d. If the angle d falls short of the circle's centre, lend it the extra space, then square it by motion and take back what was lent.
Cubing the sphere with equal pyramids
Once the surface of the sphere has been squared, the square is divided into any number of equal small squares. Each small square serves as the base of a pyramid whose axis equals the semidiameter of the sphere to be cubed, all the pyramids being equal.
Quadrature of a circular segment by motion
Under the heading 'On arc and chord', a segment of a circle is squared by means of motion that straightens out the curved periphery of the segment. A subdivided cube and a circle broken into triangles accompany the constructions.