### Propositions

Proposition 1.
Similar polygons inscribed in circles are to one another as the squares on their diameters.

Proposition 2.
Circles are to one another as the squares on their diameters.

Lemma for XII.2.

Proposition 3.
Any pyramid with 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 half of the whole pyramid.

Proposition 4.
If there are two pyramids of the same height with triangular bases, and each of them is divided into two pyramids equal and similar to one another and similar to the whole, and into two equal prisms, then the base of the one pyramid is to the base of the other pyramid as all the prisms in the one pyramid are to all the prisms, being equal in multitude, in the other pyramid.

Lemma for XII.4.

Proposition 5.
Pyramids of the same height with triangular bases are to one another as their bases.

Proposition 6.
Pyramids of the same height with polygonal bases are to one another as their bases.

Proposition 7.
Any prism with a triangular base is divided into three pyramids equal to one another with triangular bases.

Corollary. Any pyramid is a third part of the prism with the same base and equal height.

Proposition 8.
Similar pyramids with triangular bases are in triplicate ratio of their corresponding sides.

Corollary. Similar pyramids with polygonal bases are also to one another in triplicate ratio of their corresponding sides.

Proposition 9.
In equal pyramids with triangular bases the bases are reciprocally proportional to the heights; and those pyramids are equal in which the bases are reciprocally proportional to the heights.

Proposition 10.
Any cone is a third part of the cylinder with the same base and equal height.

Proposition 11.
Cones and cylinders of the same height are to one another as their bases.

Proposition 12.
Similar cones and cylinders are to one another in triplicate ratio of the diameters of their bases.

Proposition 13.
If a cylinder is cut by a plane parallel to its opposite planes, then the cylinder is to the cylinder as the axis is to the axis.

Proposition 14.
Cones and cylinders on equal bases are to one another as their heights.

Proposition 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 16.
Given two circles about the same center, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.

Proposition 17.
Given two spheres about the same center, to inscribe in the greater sphere a polyhedral solid which does not touch the lesser sphere at its surface.

Corollary to XII.17.

Proposition 18.
Spheres are to one another in triplicate ratio of their respective diameters.