In his work on the Megalithic Rings of the British Isles and France, Alexander Thom gives a number of constructions for the non-elliptical sites (the so-called egg-shaped rings and flattened circles). This paper describes two independent geometrical techniques (the Polygonal Method, the Shadow Calendar Method), each of which give the shapes provided by Thom's constructions. Neither of these methods requires the sophistication of Thom's proposed Neolithic technological culture. Although each is proposed as a possible interpretation of Megalithic rings -- the author deliberately produced two independent methods to stress the dangers of the prevalent philosophy my theory fits the observed facts -- therefore I must be correct! -- AATA