**TREATISES**

There are nine known works of
Archimedes in existence. The last
was discovered in a 10^{th} century manuscript in 1906.

**On the Sphere and Cylinder**

Thought to be his first book, this work shows that the surface area of a
sphere is four times that of a great circle (a line of shortest distance on a
sphere). He finds a way to calculate the area of any segment of a
sphere. This book also discussing
one of his more important discoveries: that
the volume of a sphere is two-thirds the volume of a circumscribed cylinder, and
that the surface of the sphere is two-thirds the surface of the cylinder.
These topics brought him close to inventing calculus and were very useful
to Newton and Leibniz in the seventeenth century.

**On the Measurement of the
Circle**

*
*In this work Archimedes made
discoveries regarding the value of pi. See
Archimedes' Constant.

**On the Equilibrium of Planes**

*
*In this treatise Archimedes discusses
fundamental principles of mathematics using geometric properties.
He discovered important theorems regarding the center of gravity of plane
figures and proved the law of the lever. It
is divided into two books. The
first contains discussions of the center of gravity of a triangle,
parallelogram, and a trapezium. The
second book considers the center of gravity of a segment of a parabola.

**On Conoids and Spheroids**

Archimedes investigates the shapes
produced by rotating parabolas, hyperbolas, spheres, and ellipses.

**On Spiral**

Archimedes’ defines a spiral and examines its properties connecting the length of its radius vector with the angles through which it revolves. He discusses the tangents of the spiral and finds the area of portions of it.

**On the Quadrature of the
Parabola**

In this volume he finds the area of a
segment of a parabola cut off by any cord.

*Sandreckoner*

Thought to be one of his last volumes,
Sandreckoner contains his famous estimation that about 10 to the 63^{rd}
power grains of sand would fill up the universe.
He devised a number system that enabled him to express numbers up to 8 x
10^63. This work is important to historians because he had to
determine the dimensions of the universe in order to calculate the number of
grains of sand it could contain. He
used theories of several philosophers of the time, including his father, to make
the projection. Sandreckoner is
said to be the beginning of the logarithmic system.

**On Floating Bodies**

In this treatise Archimedes discusses his
famous law of buoyancy. He
explains why a small stone sinks because it weighs more than the tiny amount of
water it displaces but a large ship is buoyed up by the huge weight of water it
displaces. The principle explains why things float and it is one of the
founding principles of hydrostatics.

**On the Method of Mechanical
Theorems**

This is the treatise in which Archimedes explains his tactics using geometry to find the results he discusses in his other treatises. Thomas Heath (1861-1940), an expert on the history of mathematics, wrote of the treatises:

*The
treatises are, without exception, monuments of mathematical exposition; the
gradual revelation of the plan of attack, the masterly ordering of the
propositions, the stern elimination of everything not immediately relevant to
the purpose, the finish of the whole, are so impressive in their perfection as
to create a feeling akin to awe in the mind of the reader.*

[ORIGIN] [LIFE] [TREATISES] [ARCHIMEDES' CONSTANT] [ARCHIMEDES' PRINCIPLE]

[ARCHIMEDES' SCREW] [THE CATTLE PROBLEM] [DEATH]