The Wedge Cut and the Pyramid as One-Sixth of the Solid
A rule relating any section of the wedge to the third of the wedge and the sixth of the cube
Continuing the study of the wedge (conio), the page gives a sure rule that whatever proportion a cut part bears to its remainder, it also bears to the third of the wedge and the sixth of the cube. Two prism diagrams support a proof that the pyramid a b e d is one-sixth of the body a b c g i h. Leonardo argues from halving the solid between equidistant faces and from the equality of pyramids on equal bases lying between parallel lines. Letter-labels a through i key the two figures.
On this page
A sure rule linking the cut to the third and the sixth
Since the cut of the wedge in a b c is proved to be the third of the wedge, and half of the cut c e carries half of that third, a sure rule follows: whatever proportion any cut part has with its remainder, the same it has with the third of the wedge and with the sixth of the cube.
The pyramid a b e d as one-sixth of the body
The pyramid a b e d is one-sixth of the body a b c g i h. Taking half the body between the equidistant surface a b c and base e f d, the part a b c d is one-third of that half and so one-sixth of the whole. Pyramids on the common base a b d between parallel lines and of equal height, shown by the sides d c and d e, are equal to one another.