Archimedes : biography
circa 287 BC – circa 212 BC
- On Spirals
- This work of 28 propositions is also addressed to Dositheus. The treatise defines what is now called the Archimedean spiral. It is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity. Equivalently, in polar coordinates ( ) it can be described by the equation
- , r=a+btheta
- with real numbers and . This is an early example of a mechanical curve (a curve traced by a moving point) considered by a Greek mathematician.
- On the Sphere and the Cylinder (two volumes)
- In this treatise addressed to Dositheus, Archimedes obtains the result of which he was most proud, namely the relationship between a sphere and a circumscribed cylinder of the same height and diameter. The volume is 3 for the sphere, and 23 for the cylinder. The surface area is 42 for the sphere, and 62 for the cylinder (including its two bases), where is the radius of the sphere and cylinder. The sphere has a volume that of the circumscribed cylinder. Similarly, the sphere has an area that of the cylinder (including the bases). A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request.
- On Conoids and Spheroids
- This is a work in 32 propositions addressed to Dositheus. In this treatise Archimedes calculates the areas and volumes of sections of cones, spheres, and paraboloids.
- On Floating Bodies (two volumes)
- In the first part of this treatise, Archimedes spells out the law of equilibrium of fluids, and proves that water will adopt a spherical form around a center of gravity. This may have been an attempt at explaining the theory of contemporary Greek astronomers such as Eratosthenes that the Earth is round. The fluids described by Archimedes are not , since he assumes the existence of a point towards which all things fall in order to derive the spherical shape.
- In the second part, 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, similar to the way that icebergs float. Archimedes’ principle of buoyancy is given in the work, stated as follows:
- The Quadrature of the Parabola
- In this work of 24 propositions addressed to Dositheus, Archimedes proves by two methods that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He achieves this by calculating the value of a geometric series that sums to infinity with the ratio .
- (O)stomachion
- This is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest. Archimedes calculates the areas of the 14 pieces which can be assembled to form a square. Research published by Dr. Reviel Netz of Stanford University in 2003 argued that Archimedes was attempting to determine how many ways the pieces could be assembled into the shape of a square. Dr. Netz calculates that the pieces can be made into a square 17,152 ways. The number of arrangements is 536 when solutions that are equivalent by rotation and reflection have been excluded. The puzzle represents an example of an early problem in combinatorics.
- The origin of the puzzle’s name is unclear, and it has been suggested that it is taken from the Ancient Greek word for throat or gullet, stomachos (). Ausonius refers to the puzzle as Ostomachion, a Greek compound word formed from the roots of (osteon, bone) and (machē – fight). The puzzle is also known as the Loculus of Archimedes or Archimedes’ Box.
- Archimedes’ cattle problem
- This work was discovered by Gotthold Ephraim Lessing in a Greek manuscript consisting of a poem of 44 lines, in the Herzog August Library in Wolfenbüttel, Germany in 1773. It is addressed to Eratosthenes and the mathematicians in Alexandria. Archimedes challenges them to count the numbers of cattle in the Herd of the Sun by solving a number of simultaneous Diophantine equations. There is a more difficult version of the problem in which some of the answers are required to be square numbers. This version of the problem was first solved by A. AmthorKrumbiegel, B. and Amthor, A. Das Problema Bovinum des Archimedes, Historisch-literarische Abteilung der Zeitschrift Für Mathematik und Physik 25 (1880) pp. 121–136, 153–171. in 1880, and the answer is a very large number, approximately 7.760271.