Hippocrates of Chios (470 B.C.) was the first to investigate areas bounded by curves and to bring into prominence the problems of squaring the circle and duplication of the cube. According to Philoponus, the Athenians were suffering from a severe plague of typhoid fever in 430 B.C., and were told by the oracle at Delos that Apollo required an altar in the form of a cube twice the size of the one existing.
Heat the butter or margarine, stir in the cornflour
or flour. Cook gently for several minutes. Add the
water, stock cube and milk and bring to the boil.
Cook until thickened. Add the sweet corn and heat
for a few minutes. Season well and garnish with
chopped chives or parsley.
Next to the squaring of the circle, the most famous problems of antiquity are the Delium problem of the duplication of the cube and the trisection of an arbitrary angle. Granting the preceding general theorem, these are easily shown to be impossible when restricted to the straight edge and compass.
The duplication of the cube requires the determination of the edge of a cube x, such that its cube shall be twice the volume of a given unit. That is x3 = 2. This equation is irreducible, since otherwise v2 would be rational. Moreover, the equation is a cubic and its degree is not of the form 2n. Hence the solution is in general impossible.
Hippocrates of Chios (470 B.C.) was the first to investigate areas bounded by curves and to bring into prominence the problems of squaring the circle and duplication of the cube. According to Philoponus, the Athenians were suffering from a severe plague of typhoid fever in 430 B.C., and were told by the oracle at Delos that Apollo required an altar in the form of a cube twice the size of the one existing.