- Propositions (18)

**Proposition 1.**- If a straight line is cut in extreme and mean ratio, then the square on the greater segment added to the half of the whole is five times the square on the half.
**Proposition 2.**- If the square on a straight line is five times the square on a segment on it, then, when the double of the said segment is cut in extreme and mean ratio, the greater segment is the remaining part
of the original straight line.
**Lemma**for XIII.2. **Proposition 3.**- If a straight line is cut in extreme and mean ratio, then the square on the sum of the lesser segment and the half of the greater segment is five times the square on the half of the greater segment.
**Proposition 4.**- If a straight line is cut in extreme and mean ratio, then the sum of the squares on the whole and on the lesser segment is triple the square on the greater segment.
**Proposition 5.**- If a straight line is cut in extreme and mean ratio, and a straight line equal to the greater segment is added to it, then the whole straight line has been cut in extreme and mean ratio, and the original straight line is the greater segment.
**Proposition 6.**- If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome.
**Proposition 7.**- If three angles of an equilateral pentagon, taken either in order or not in order, are equal, then the pentagon is equiangular.
**Proposition 8.**- If in an equilateral and equiangular pentagon straight lines subtend two angles are taken in order, then they cut one another in extreme and mean ratio, and their greater segments equal the side of the pentagon.
**Proposition 9.**- If the side of the hexagon and that of the decagon inscribed in the same circle are added together, then the whole straight line has been cut in extreme and mean ratio, and its greater segment is the side of the hexagon.
**Proposition 10.**- If an equilateral pentagon is inscribed ina circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle.
**Proposition 11.**- If an equilateral pentagon is inscribed in a circle which has its diameter rational, then the side of the pentagon is the irrational straight line called minor.
**Proposition 12.**- If an equilateral triangle is inscribed in a circle, then the square on the side of the triangle is triple the square on the radius of the circle.
**Proposition 13.**- To construct a pyramid, to comprehend it in a given sphere; and to prove that the square on the diameter of the sphere is one and a half times the square on the side of the pyramid.
**Lemma**for XIII.13. **Proposition 14.**- To construct an octahedron and comprehend it in a sphere, as in the preceding case; and to prove that the square on the diameter of the sphere is double the square on the side of the octahedron.
**Proposition 15.**- To construct a cube and comprehend it in a sphere, like the pyramid; and to prove that the square on the diameter of the sphere is triple the square on the side of the cube.
**Proposition 16.**- To construct an icosahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the icosahedron is the irrational straight line called minor.
**Corollary.**The square on the diameter of the sphere is five times the square on the radius of the circle from which the icosahedron has been described, and the diameter of the sphere is composed of the side of the hexagon and two of the sides of the decagon inscribed in the same circle. **Proposition 17.**- To construct a dodecahedron and comprehend it in a sphere, like the aforesaid figures; and to prove that the square on the side of the dodecahedron is the irrational straight line called apotome.
**Corollary.**When the side of the cube is cut in extreme and mean ratio, the greater segment is the side of the dodecahedron. **Proposition 18.**- To set out the sides of the five figures and compare them with one another.
**Remark.**- No other figure, besides the said five figures, can be constructed by equilateral and equiangular figures equal to one another.
**Lemma.**The angle of the equilateral and equiangular pentagon is a right angle and a fifth.