From what I can gather at Perseus, it is now the scholarly contention that Archimedes used, or at least had knowledge of, the Antikythera Mechanism, the world's oldest computer -- http://en.wikipedia.org/wiki/Antikythera_mechanism -- and possibly used it in constructing his planetarium.
It is well known that Archimedes used the geometric algebra:Geometric algebra could be used to solve quadratic equations (x^2 + x + 1 = 0) in a similar way that modern algebra is used today. However, the complicated nature of geometric algebra prevented it from being used to solve more complex polynomials (x^3 + x^2 + x + 1 = 0).
[http://vanth.perseus.tufts.edu/GreekScience/Students/Mike/geometry.html] Greek Geometrical AlgebraThe Greeks of the classical period, who did not recognize the existence of irrational numbers, avoided the problem thus created by representing quantities as geometrical magnitudes. Various algebraic identities and constructions equivalent to the solution of quadratic equations were expressed and proven in geometric form. In content there was little beyond what the Babylonians had done, and because of its form geometrical algebra was of little practical value. This approach retarded progress in algebra for several centuries. The significant achievement was in applying deductive reasoning and describing general procedures.
[http://www.ucs.louisiana.edu/~sxw8045/history.htm] http://mathworld.wolfram.com/ArchimedesCattleProblem.html also http://www.bookrags.com/wiki/Archimedes * Archimedes' Cattle ProblemArchimedes 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 [please note; EY] 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
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