# Archimedes Nine Surviving Treatises

• On the Sphere and Cylinder (in two books)

shows the surface area of any sphere is 4 pi r2, and the volume of a sphere is two-thirds that of the cylinder in which it is inscribed, V = 4/3 pi r3

• Measurement of the Circle

shows that pi, the ratio of the circumference to the diameter of a circle, is between
3 10/70 and 3 10/71

• On Conoids and Spheroids

finds the volumes of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis

• On Spirals

develops many properties of tangents to the spiral of Archimedes

• Centers of Gravity of Planes (in two books)

finds the centers of gravity of various plane figures and conics, and establishes the "law of the lever"

finds, first by "mechanical" means (Archimedes' "Method") and then by rigorous geometry, the area of any segment of a parabola

• The Sand Reckoner

attempts to  remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number - the number of grains of sand that it would take to fill the whole of the universe by creating a place-value system of notation, with a base of 100,000,000 (and contains the most detailed surviving description of the heliocentric system of Aristarchus of Samos)

• Method Concerning Mechanical Theorems

describes the process of discovery in mathematics

• On Floating Bodies (in two books)

finds the positions that various solids will assume when floating in a fluid, and establishes Archimedes' principle (that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object)