  Menaechmus (ca 350 B.C.) was another student in Plato’s academy. He discovered the conics: the ellipse, hyperbola, and parabola. These arise as points of intersection of planes passing through right circular cones (or pairs of cones). Plato insisted that geometric prob- lems should have straight edge and compass constructions. It was shown much later (in 1837 by Pierre Wantzel) that it is not possible to double the cube by ruler and compass constructions only. But Menaechmus described a simple way of using his conics to double a cube: √3 3 (i) Show that if we want to double a cube, it suffices to find 2, that is to solve x = 2. (ii) Show (i) arises by finding the intersection of the parabola y = 1x2 with the hyperbola 2 xy = 1.

Question

Menaechmus (ca 350 B.C.) was another student in Plato’s academy. He discovered the conics: the ellipse, hyperbola, and parabola. These arise as points of intersection of planes passing through right circular cones (or pairs of cones). Plato insisted that geometric prob- lems should have straight edge and compass constructions. It was shown much later (in 1837 by Pierre Wantzel) that it is not possible to double the cube by ruler and compass constructions only. But Menaechmus described a simple way of using his conics to double a cube: √3 3 (i) Show that if we want to double a cube, it suffices to find 2, that is to solve x = 2. (ii) Show (i) arises by finding the intersection of the parabola y = 1x2 with the hyperbola 2 xy = 1.

Step 1

Aspects of the classical problem of doubling the cube: algebraic and geometic connections

Step 2

Historically, the problem of doubling a cube is stated as follows: Cube A is given edges a and volume a^3. Required to

"double the cube A": that is , to construct cube B whose volume is twice that of the given cube A.

Step 3

In this problem , we are only concerned with the aspects of alg...

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