TREATISES

 

 

There are nine known works of Archimedes in existence.  The last was discovered in a 10th 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

            Archimedesí 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 63rd 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.

 

We know that more works of Archimedes once existed.  Archimedes himself and several other philosophers make reference to them.  They explored topics including semi-regular polyhedra and the number system he used in Sandreckoner.

 

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

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

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