Almost all of the different types of megalithic building  were evolved in the fifth millennium (5,000-4,000 BC), in the area around Carnac on southern Brittany's Atlantic coast. This includes the stone circles later extensively built in the British Isles. When Alexander Thom surveyed these, between 1934 and 1978, he found them to be remarkably technical constructions, involving sophisticated geometrical ideas. It was only in the mid-seventies, when Thom came to Carnac, that the same geometries were found applied within Carnac's stone circles though at that time it was not known that Carnac's monuments preceded those of Britain by at least a thousand years.
 Megalithic building types include standing stones, stone circles, stone rows, dolmen, chambered and other cairns.
After an initial enthusiasm for Thom's work, in the late 1960s and early 1970s, British archaeologists chose, with very few exceptions, to refute the entire notion that the Neolithic could have been constructing such technical geometries which, as far as our History would have it, would only have become possible after the development (over two thousand miles away in the ancient near east) of a functional mathematics which culminated in Euclid's classical work on analytic geometry, Elements. Thom's use of geometry therefore seemed anachronistic to archaeologists and to accept it would have required a revolution in our thinking about the megalithic; for this there was little appetite. It was easier to falsify Thom's hypothesis with new work concluding that, for example, "stone circles were distorted so that the audience could see all the rites; and the principals could occupy visually focal positions facing the spectators.", clearly indicating the current "comfort zone" within archaeology in which unquestioned ideas about superstitious rites are used to supersede Thom's accurate and well founded proposals, of a megalithic technical capability, as being overly technical. The problem with inventing such ancient rites as being the primary purpose for stone circle construction is that, whilst refuting Thom's proposal, it definitely cannot be proved, cannot be disproved; Talk of rites as being the reason for stone circles is not delivering a scientific theory and Thom's proposals are not disproved by such ideas.
Figure Thom's site plans of two of Britain's finest surviving Flattened Circles, Castle Rigg (Type A) and Long Meg (Type B). Castle Rigg points (within a degree) to Long Meg, on a bearing which follows the diagonal of a two by one (east by north) rectangle, as if (despite some Lake District mountains in between) the two sites were related when built and hence contemporaneous.
Unlike most of his detractors, Thom physically engaged with stone circles, by surveying them, and through this activity he was to create the first and only extensive corpus of stone circle site plans. Through this he left a vitally important legacy by, at the very least, preserving their layout against natural and man-made degradation. The geometrical overlays and typology found within Thom's site plans have been dismissed as unlikely, on spurious technical grounds [*], usually by people with insufficient technical background in the technical issues within his work. Despite this, Thom's later work in Carnac has proven critical in providing further alternative explanations as to how the megalithic actually constructed these stone circle geometries without Euclidian geometrical methods, using instead the system of multiple squares found to be in use in the megalithic structures around Carnac; this in the late 1970's [note on AAK and Howard Crowhurst], and after Thom's surveying seasons there.
Such a system of squares avoids the arcing of ropes used by Thom to explain the construction of different stone designs. Instead, a grid of squares can establish the few key points on the perimeter of a flattened circle. This would eliminate one of the main objections to Thom's stone circle geometries: that a Euclidian geometrical process, of lines and arcs, is anachronistic in the context of a megalithic three to four thousand years before Euclid.
Figure The geometries of Flattened Circles (left to right) called Types A, B and D
In the case of the Type A (flattened) stone circles proposed by Thom, I demonstrate below that accepting Thom's identification of its geometry is a necessary stepping stone to understanding how this could be achieved by a pre-arithmetic megalithic of the fifth millennium BC. By the Middle Kingdom, the Egyptians had put stylus to papyrus, to describe their mathematics in the document now called the Rhind Manuscript. This recorded a system of geometry based around pre-Ptolemaic ideas, that included the use of a grid of multiple squares.
At Carnac, the angular extremes of sunrise and sunset, on the horizon during the year, followed the lesser angle of a 3-4-5 triangle whilst in the Rhind Manuscript one finds a "canevas" [*] or grid-based diagram, in which both of the acute angles of this triangle are shown to be generated by the summed diagonal angles of either; two double squares or two triple squares. The resulting grid is then 14 squares by 14 squares, and this is exactly the grid upon which the Type A stone circles are most easily constructed if one excludes the use of ropes, stakes and measuring rods to construct these designs.
Figure of Rhind diagram showing evolution of a 3-4-5 triangle within a 14 by 14 grid of squares
However, such a use of squares to construct a stone circle geometry immediately raises the question of the side length used, since they all need to be identical and so an ability to create identical lengths almost certainly points to an accurate system of measures, a metrology. This leads us into another quite bitter dispute, concerning Alexander Thom's ideas for the existence of a megalithic yard as primary unit of measure, maintained accurately by the megalithic builders throughout the British Isles and Brittany. Unfortunately, Thom did not know enough about historical metrology to see that the megalithic yard might well have been accompanied by systematic variations applied to its length or indeed, other measures might also have evolved. His proposal of an accurate megalithic yard, like that of exact stone circle geometries, therefore came to be rejected since his detractors, who also knew very little about historical metrology [*], could point to cases where, in fact, the actual presence of other measures had muddied any proof of there had only been a singular measure in megalithic Britain.
*[Historical metrology is a scattered remnant of the metrological system employed within the British stone circles and also within the Egyptian pyramids. It is this latter application of metrology in the ancient near east which spread metrology, though such an idea has also been opposed by archaeologists working in the near east.]