Exact Doubling the Cube with Straightedge and Compass by Euclidean Geometry

No great theory lasts forever in science, but only specific research and discoveries continuously complement each other. There have been 3 classical problems remaining from ancient Greek mathematics, which are extremely influential in the development of geometry. They are Trisecting An Angle, Squaring The Circle, and Doubling The Cube problems. The “Doubling The Cube” problem is stated: Using only a straightedge and a compass, is it possible to construct a cube whose volume is double the volume a³ of a given cube?. From the oldest mathematical documents known up to today's mathematics, the "Doubling The Cube” problem has interested professional & non-professional mathematicians for millenniums. The technique “ANALYSIS” is adopted to solve accurately and exactly the “Doubling The Cube” problem with only a straightedge & compass by Euclidean Geometry, and does not change any premise of the problem. Upstream from this method of exact “Doubling The Cube”, one can deduce an equivalence to get a new Mathematical challenge "Halving The Cube” (i.e. dividing a given cube into 2 smaller equal cubes, using only a straightedge & a compass. This independent research shows an exact precision and accurate solution for the ancient Greek challenge – “Doubling The Cube” using a straightedge and a compass only. Mathematics tools and propositions used in this solution are all in Euclidean Geometry and algebraic geometry. The methodology of the solution includes geometrical methods to arrange the given cube and its double volume cube into a concentric position of which side x of the double cube can be calculated accurately by algebra then in terms of geometrical length the side is constructive. Finally, use the straightedge and the compass to construct the double-volume cube.