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Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.
It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics.
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a ...
A straightedge or straight edge is a tool used for drawing straight lines, or checking their straightness. If it has equally spaced markings along its length, it is usually called a ruler . Straightedges are used in the automotive service and machining industry to check the flatness of machined mating surfaces.
To draw the parallel (h) to a diameter g through any given point P. Chose auxiliary point C anywhere on the straight line through B and P outside of BP. (Steiner) In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules.
Bisection of arbitrary angles has long been solved.. Using only an unmarked straightedge and a compass, Greek mathematicians found means to divide a line into an arbitrary set of equal segments, to draw parallel lines, to bisect angles, to construct many polygons, and to construct squares of equal or twice the area of a given polygon.
Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship with other human activities. Major themes that are dealt with in philosophy of mathematics include: Reality: The question is whether mathematics is a pure product of human mind or whether it has some reality by itself. Logic and ...
Projective geometry can also be seen as a geometry of constructions with a straight-edge alone, excluding compass constructions, common in straightedge and compass constructions. [2] As such, there are no circles, no angles, no measurements, no parallels, and no concept of intermediacy (or "betweenness"). [3]