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Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume. [5] It seems that this is not the original method Archimedes used to derive this result, but the best formal argument available to him in the Greek mathematical tradition.
In ancient Greek geometry, the Ostomachion, also known as loculus Archimedius (from Latin 'Archimedes' box') or syntomachion, is a mathematical treatise attributed to Archimedes. This work has survived fragmentarily in an Arabic version and a copy, the Archimedes Palimpsest , of the original ancient Greek text made in Byzantine times.
Proof: [1]: 15–18 Let D be the midpoint of AC. Construct a line segment JB through D, where the distance from J to D is equal to the distance from B to D. We will think of the segment JB as a "lever" with D as its fulcrum. [3] As Archimedes had previously shown, the center of mass of the triangle is at the point I on the "lever" where DI :DB ...
Also known as Loculus of Archimedes or Archimedes' Box, [87] this is a dissection puzzle similar to a Tangram, and the treatise describing it was found in more complete form in the Archimedes Palimpsest.
The Archimedes Palimpsest is a parchment codex palimpsest, originally a Byzantine Greek copy of a compilation of Archimedes and other authors. It contains two works of Archimedes that were thought to have been lost (the Ostomachion and the Method of Mechanical Theorems ) and the only surviving original Greek edition of his work On Floating ...
In this setting, an ordered field K is Archimedean precisely when the following statement, called the axiom of Archimedes, holds: "Let x {\displaystyle x} be any element of K {\displaystyle K} . Then there exists a natural number n {\displaystyle n} such that n > x {\displaystyle n>x} ."
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Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity, with weighted segments of a parabola and a triangle suspended along the arms of a lever at specific distances from the fulcrum. [4]