Euclid's Elements

Similar prisms which have triangular bases are divided into similar pyramids, equal in number: and a prism to a prism has a triple ratio of that which a homologous side has to a homologous side. 217.
Similar prisms which have multi-angled bases are divided into similar prisms having triangular bases, equal in number, and homologous to the wholes: and a prism to a prism has a triple ratio of that which a homologous side has to a homologous side. 217.b
Of equal pyramids having multi-angled bases, the bases correspond to the heights on the opposite side: and those pyramids having multi-angled bases whose bases correspond to the heights on the opposite side are equal. 218.b
Of all equal prisms, the bases correspond to the heights on the opposite side, and those prisms whose bases correspond to the heights on the opposite side are equal. 219
Every cone, whether right or scalene, is the third part of a cylinder, whether right or scalene, which has the same base and the same height. 220
All similar cones and cylinders are in the triple ratio of the diameters which are in their bases to each other. 222
If a scalene cylinder is cut by a plane parallel to the opposite planes, it will be as the cylinder is to the cylinder, so is the axis to the axis. 223.b
If any cylinder is cut by a plane parallel to the bases, it will be as the cylinder is to the cylinder, so is the height of the cylinder to the height of the cylinder.
Of all equal cones and cylinders, the bases correspond to the heights on the opposite side, and those cones and cylinders whose bases correspond to the heights on the opposite side are equal to each other. 224
All cylinders and cones have a ratio between them compounded from the ratio of the bases and the ratio of the heights. 224.b

IN THE THIRTEENTH BOOK.

Proposition 1 is demonstrated otherwise. 229.b

A straight line being given, cut in extreme and mean ratio, each of its segments will be given.

If a rational straight line, commensurable in length with an assigned rational line, is cut in extreme and mean ratio, its greater segment will be the fifth apotome a type of irrational line segment, and the lesser will be the first apotome.
Given the greater segment, to find the whole straight line which is cut in extreme and mean ratio.
Given the greater segment of a straight line which is cut in extreme and mean ratio, both the lesser segment and the whole line are given. 230
Proposition 2 is demonstrated otherwise. 230.b
Given the lesser segment, to find the whole straight line which is cut in extreme and mean ratio. 231.b
Given the lesser segment of a straight line which is cut in extreme and mean ratio, both the greater segment and the whole line are given.
If a straight line is cut in extreme and mean ratio, and a line equal to the lesser segment is cut off from the greater segment, it will also be cut in extreme and mean ratio, and the greater segment will be the straight line that was cut off. 232.b
If the greater segment of a straight line cut in extreme and mean ratio is rational, and commensurable in length with an assigned rational line; the lesser segment will be the fifth apotome, and the whole will be the fifth binomial a type of irrational line segment. 233
If the lesser segment of a straight line cut in extreme and mean ratio is rational, and commensurable in length with an assigned rational line, the greater segment will be the fifth binomial, and the whole will be the first binomial. 233.b
If the side of a hexagon is cut in extreme and mean ratio, its greater segment will be the side of a decagon. 234.b
If an equilateral decagon is described in a circle having a rational diameter, the side of the decagon will be the fifth apotome. 235
If the side of an equilateral decagon described in a circle is rational, the diameter of the circle will be the fifth binomial. 235