The Euclid's Elements of Geometry Book XIII: Regular solids Propositions
The Euclid's Elements of Geometry
Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.
Book XIII: Regular solids
Propositions
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.
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.
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.
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.
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.
If a rational straight line is cut in extreme and mean ratio, then each of the
segments is the irrational straight line called apotome.
If three angles of an equilateral pentagon, taken either in order or not in
order, are equal, then the pentagon is equiangular.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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, thegreater segment is the side of the dodecahedron.
To set out the sides of the five figures and compare them with one another.