Chronos
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Member # 2051
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posted 12-21-2005 12:15 PM
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Writings by Archimedes
On the Equilibrium of Planes (2 volumes)
This scroll explains the law of the lever and uses it to calculate the areas and centers of gravity of various geometric figures.
On Spirals
In this scroll, Archimedes defines what is now called Archimedes' spiral. This is the first mechanical curve (i.e., traced by a moving point) ever considered by a Greek mathematician. Using this curve, he was able to square the circle.
On the Sphere and The Cylinder
In this scroll Archimedes obtains the result he was most proud of: that the area and volume of a sphere are in the same relationship to the area and volume of the circumscribed straight cylinder.
On Conoids and Spheroids
In this scroll Archimedes calculates the areas and volumes of sections of cones, spheres and paraboloids.
On Floating Bodies (2 volumes)
In the first part of this scroll, Archimedes spells out the law of equilibrium of fluids, and proves that water around a center of gravity will adopt a spherical form. This is probably an attempt at explaining the observation made by Greek astronomers that the Earth is round. Note that his fluids are not self-gravitating: he assumes the existence of a point towards which all things fall and derives the spherical shape. One is led to wonder what he would have done had he struck upon the idea of universal gravitation.
In the second part, a veritable tour-de-force, he calculates the equilibrium positions of sections of paraboloids. This was probably an idealization of the shapes of ships' hulls. Some of his sections float with the base under water and the summit above water, which is reminiscent of the way icebergs float, although Archimedes probably was not thinking of this application.
The Quadrature of the Parabola
In this scroll, Archimedes calculates the area of a segment of a parabola (the figure delimited by a parabola and a secant line not necessarily perpendicular to the axis). The final answer is obtained by triangulating the area and summing the geometric series with ratio 1/4.
Stomachion
This is a Greek puzzle similar to Tangram. In this scroll, Archimedes calculates the areas of the various pieces. This may be the first reference we have to this game. Recent discoveries indicate that Archimedes was attempting to determine how many ways the strips of paper could be assembled into the shape of a square. This is possibly the first use of combinatorics to solve a problem.
Archimedes' Cattle Problem
Archimedes wrote a letter to the scholars in the Library of Alexandria, who apparently had downplayed the importance of Archimedes' works. In these letters, he dares them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations, some of them quadratic (in the more complicated version). This problem is one of the famous problems solved with the aid of a computer. The solution is a very large number, approximately 7.760271 × 10206544 (See the external links to the Cattle Problem.)
The Sand Reckoner
In this scroll, Archimedes counts the number of grains of sand fitting inside the universe. This book mentions Aristarchus' theory of the solar system, contemporary ideas about the size of the Earth and the distance between various celestial bodies. From the introductory letter we also learn that Archimedes' father was an astronomer.
"The Method"
In this work, which was unknown in the Middle Ages, but the importance of which was realised after its discovery, Archimedes pioneered the use of infinitesimals, showing how breaking up a figure in an infinite number of infinitely small parts could be used to determine its area or volume. Archimedes did probably consider these methods not mathematically precise, and he used these methods to find at least some of the areas or volumes he sought, and then used the more traditional method of exhaustion to prove them. This particular work is found in what is called the Archimedes Palimpsest. Some details can be found at how Archimedes used infinitesimals.
[ 12-21-2005, 12:16 PM: Message edited by: Chronos ]
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