Archimedes

The surface of any sphere is four times the area of its greatest circle.

The volume of any sphere is four times the volume of the cone whose base is a great circle of the sphere and whose height is equal to the radius of the sphere.

The volume of any segment of a sphere is equal to the volume of a cone whose base is the base of the segment and whose height is equal to the height of the segment, plus the volume of a cylinder whose base is the base of the segment and whose height is equal to the radius of the sphere minus the height of the segment.

The spiral of Archimedes is a curve such that the distance from the origin to any point on the curve is proportional to the angle which the line joining the origin to that point makes with a fixed line.

The area of the first revolution of the spiral is one-third of the area of the circle whose radius is the distance from the origin to the point where the spiral completes its first revolution.

The tangent to the spiral at any point makes an angle with the radius vector to that point such that the tangent of the angle is equal to the ratio of the radius vector to the arc of the circle whose radius is the radius vector and whose angle is the angle of the spiral.

The number of grains of sand that could be contained in the universe is not infinite.

The universe is a sphere whose center is the center of the Earth and whose radius is the distance from the center of the Earth to the fixed stars.

The diameter of the sun is 30 times the diameter of the Earth.

The diameter of the moon is 1/9th the diameter of the Earth.

The volume of a cylinder is three times the volume of a cone with the same base and height.

The surface area of a cone, excluding its base, is equal to the area of a circle whose radius is the slant height of the cone and whose circumference is the circumference of the base of the cone.

The center of gravity of any segment of a parabola is on the diameter of the segment, and divides the diameter in the ratio 3:2, the part nearer the vertex being the greater.

The center of gravity of any segment of a sphere is on the axis of the segment, and divides the axis in a certain ratio.

The center of gravity of any paraboloid of revolution is on its axis, and divides the axis in the ratio 2:1, the part nearer the vertex being the greater.

The volume of any segment of a paraboloid of revolution cut off by a plane perpendicular to the axis is one and a half times the volume of the cone which has the same base and the same height.

The volume of any segment of a hyperboloid of revolution cut off by a plane perpendicular to the axis is equal to the volume of a cone whose base is the base of the segment and whose height is equal to the height of the segment, plus the volume of a cylinder whose base is the base of the segment and whose height is equal to the height of the segment multiplied by the ratio of the square of the radius of the base to the square of the semi-axis of the hyperbola.

The volume of any segment of an ellipsoid of revolution cut off by a plane perpendicular to the axis is equal to the volume of a cone whose base is the base of the segment and whose height is equal to the height of the segment, minus the volume of a cylinder whose base is the base of the segment and whose height is equal to the height of the segment multiplied by the ratio of the square of the radius of the base to the square of the semi-axis of the ellipse.

The center of gravity of any segment of a paraboloid of revolution cut off by a plane perpendicular to the axis is on the axis of the segment, and divides the axis in the ratio 2:1, the part nearer the vertex being the greater.

The center of gravity of any segment of a hyperboloid of revolution cut off by a plane perpendicular to the axis is on the axis of the segment, and divides the axis in a certain ratio.