Edited sentence: Either analysis yields an irrational number value for the volume of an integer side cube; proving that the sum of two cubes, each having an integer valued volume, is not a cube of integer valued volume.He could have determined the volume of a cube by analyzing the number of 3,4,5 right triangles required to cover the  cube's faces,or analyzing the surface area of slices from one cube needed to be applied to the faces of the other in order to construct a third cube.His epiphany likely came upon realizing that the sum of the volumes of two polyhedrons can be treated as the sum of two cubes having corresponding volumes; and that, furthermore, the sum of two cubes is a single cube having the same net volume.However,since it would be reasonable for Fermat to look for a solid geometry theorem like Pythagoras' theorem, his note might be better interpreted as stating that he had discovered a remarkable proof that, (like the square on the hypotenuse), the volume of a cube, a tetrahedron, and any other polyhedron can also be divided into two.The argument Menaechmus used around 500 BC to arrive at the conclusion that a precise solution to the Delian problem was impossible may have been similar