Archimedes of Syracuse

Any magnitude whatever being given, and any other magnitude of the same kind, it is possible to find a multiple of the second magnitude which will exceed the first.

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

The volume of any sphere is two-thirds the volume of its circumscribing cylinder.

The area of a segment of a parabola is four-thirds of the triangle with the same base and height.

If a solid lighter than a fluid is immersed in it, it will not be submerged so as to touch the bottom of the fluid, but will be borne up into such a position that the part of the solid outside the fluid has to the part within it the same ratio as the excess of the weight of the fluid over the weight of the solid has to the weight of the solid.

There are no such things as numbers which are too large to be named.

The number of grains of sand that could be contained in the universe is less than 10^63.

The ratio of the circumference of any circle to its diameter is the same as the ratio of the area of the circle to the square on its radius.

The area of a circle is equal to the area of a right-angled triangle whose base is equal to the circumference of the circle and whose height is equal to the radius of the circle.

The center of gravity of any plane figure is the point such that if the figure is suspended from that point, it will remain in equilibrium.

Magnitudes are said to be in equilibrium when they are suspended at distances reciprocally proportional to their weights.

The center of gravity of a triangle is the point where its medians intersect.

The area of an ellipse is to the area of its circumscribing circle as the product of its semi-axes is to the square of its radius.

The volume of a 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 spiral of Archimedes is a curve such that the distance from the origin to any point on the curve is proportional to the angle through which the radius vector has turned.

The area enclosed by the first turn of an Archimedean spiral and the initial line is one-third of the area of the circle whose radius is the length of the first turn.

The method of exhaustion is a method of finding the area or volume of a figure by inscribing and circumscribing polygons or polyhedra with an increasing number of sides.

The value of pi can be approximated by taking the average of the perimeters of inscribed and circumscribed regular polygons.

The method of mechanical theorems is a method of finding the area or volume of a figure by considering it as a balance of weights.

By means of this method, we shall be able to arrive at the knowledge of some theorems in geometry by mechanical considerations.