Of Geometry the twenty fifth Booke; Of mingled ordinate Polyedra's.
1. A mingled ordinate polyedrum is a pyramidate, compounded of pyramides with their toppes meeting in the center, and their bases onely outwardly appearing.
Seeing therefore a Mingled ordinate pyramidate is thus made or compounded of pyramides the geodesy of it shall be had from the Geodesy of the pyramides compounding it: And one Base multiplyed by the number of all the bases shall make the surface of the body. And one Pyramis by the number of all the pyramides; shall make the solidity.
2 The heighth of the compounding pyramis is found by the ray of the circle circumscribed about the base, and by the semidiagony of the polyedrum.
The base of the pyramis appeareth to the eye: The heighth lieth hidde within, but it is discovered by a right angle triangle, whose base is the semidiagony or halfe diagony, the shankes the ray of the circle, and the perpendicular of the heighth. Therefore subtracting the quadrate of the ray, from the quadrate of the halfe diagony the side of the remainder, by the [9 e xij]. shall be the heighth. But the ray of the circle shall have a speciall invention, according to the kindes of the base, first of a triangular, and then next of a quinquangular.
3 A mingled ordinate polyedrum hath either a triangular, or a quinquangular base.
The division of a Polyhedron ariseth from the bases upon which it standeth.
4 If a quadrate of a triangular base be divided into three parts, the side of the third part shall be the ray of the circle circumscribed about the base.