Centre of Gravity of Pyramids and Quadrature of the Cube
The gravity centre at one-quarter of the axis; solids transformed
Two large pyramids drawn at the top, threaded with intersecting internal lines, illustrate that the centre of every pyramidal gravity lies at one-quarter of the axis toward the base: divide the axis into four equal parts and intersect two axes of the pyramid, and the crossing falls at that fourth. Below, a cube containing a parallelepiped (labelled g f) shows that the squaring of the larger cube is double that of the smaller, and, each resolved into six pyramids, their centre of natural gravity lies on the dividing line. A sequence of box and prism figures (a b, c d, r S, e p, n L) traces a smaller solid drawn from the larger and reshaped, halving a cube and squaring its side toward a proof of quadrature. A cross-reference sends the reader to page 59.
On this page
Centre of gravity of a pyramid at one-quarter of its axis
The centre of every pyramidal gravity is at the fourth of its axis toward the base. Divide the axis into four equal parts and intersect two of the pyramid's axes, and that intersection comes at the said fourth. The two pyramids at the top are drawn with their internal axes crossing.
Cube, contained parallelepiped, and doubled quadrature
The squaring of the larger cube (with parallelepiped g f) is double the squaring of the smaller cube; and each being resolved into its six pyramids, the centre of their natural gravity lies on the line that divides the smaller cube from the larger.
Extracting and reshaping the solid toward half a cube
By the seventh proposition the sided cylinder b is drawn from cube f, being half of it; placed at c, its side r is squared by the eighth and set at d; the square d is halved at n, and side n squared and set at p, whose figure is half a cube. Proving the squaring of the half-cube first makes proving the quarter easier with that triangle (figures a b, c d, r S, e p, n L).