BOOK XII. PROPOSITIONS.¶
Proposition XII.1
Similar polygons inscribed in circles are to one another as the squares on the diameters.
Proposition XII.2
Circles are to one another as the squares on the diameters.
Proposition XII.3
Any pyramid which has a triangular base is divided into two pyramids equal and similar to one another, similar to the whole and having triangular bases, and into two equal prisms; and the two prisms are greater than the half of the whole pyramid.
Proposition XII.4
If there be two pyramids of the same height which have triangular bases, and each of them be divided into two pyramids equal to one another and similar to the whole, and into two equal prisms, then, as the base of the one pyramid is to the base of the other pyramid, so will all the prisms in the one pyramid be to all the prisms, being equal in multitude, in the other pyramid.
Proposition XII.5
Pyramids which are of the same height and have triangular bases are to one another as the bases.
Proposition XII.6
Pyramids which are of the same height and have polygonal bases are to one another as the bases.
Proposition XII.7
Any prism which has a triangular base is divided into three pyramids equal to one another which have triangular bases.
Proposition XII.8
Similar pyramids which have triangular bases are in the triplicate ratio of their corresponding sides.
Proposition XII.9
In equal pyramids which have triangular bases the bases are reciprocally proportional to the heights; and those pyramids in which the bases are reciprocally proportional to the heights are equal.
Proposition XII.10
Any cone is a third part of the cylinder which has the same base with it and equal height.
Proposition XII.11
:term:`Cones <cone>` and :term:`cylinders <cylinder>` which are of the same :term:`height <height>` are to one another as their bases.
Proposition XII.12
Similar cones and cylinders are to one another in the triplicate ratio of the diameters in their bases.
Proposition XII.13
If a cylinder be cut by a plane which is parallel to its opposite planes, then, as the cylinder is to the cylinder, so will the axis be to the axis.
Proposition XII.14
Cones and cylinders which are on equal bases are to one another as their heights.
Proposition XII.15
In equal cones and cylinders the bases are reciprocally proportional to the heights; and those cones and cylinders in which the bases are reciprocally proportional to the heights are equal.
Proposition XII.16
Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.
Proposition XII.17
Given two spheres about the same centre, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.
Proposition XII.18
Spheres are to one another in the triplicate ratio of their respective diameters.