  While geometry with transcendental curves, like the Quadratrix of Hippias and
the Spiral of Archimedes, played a significant role in our modern developments
of geometry and algebra. The investigation has fallen off in the modern era
despite advancements in algebraic tooling. This paper gives a description of
the fields using modern techniques such as Galois theory while solving an open
conjecture in a 1988 paper to provide an answer to if these curves can solve
the problem of doubling the cube.
