Algebraic Properties of Euclidean Geometry with Transcendental Curves

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.