**Alexander Thom found these were of two main designs, Types A and B**, which reduced the perimeter of a circle defined by an initial radius/diameter. Flattened circles were found to be of fixed design in their geometry, unlike the stone circles that expanded their perimeter length, called "eggs" and this implies that eggs were in some way geared to flexible extension of perimeter, as were ellipses and compound circles, whilst flattened circles probably had a different function such as the extention of alignments when land surveying, or as astronomical observatories. The ratio of perimeter reduction achieved and the equivalent "pi" between the circle and its flattened counterpart, can be stated as perimeter reduction PR = "pi"/PI; The ratio between initial diameter MN and reduced diameter, on the axis of symmetry, AB, was another key invariant of these designs.

Type A: | PR = 0.9737 | "pi" = 3.0591 | AB/MN = 0.9114 |

Type B: | PR = 0.9667 | "pi" = 2.9572 | AB/MN = 0.8604 |

Type D: | PR = 0.9817 | "pi" = 3.0840 | AB/MN = 0.9343 |

[MSB, p28-9]

*Thom's Taxonomy of the Stone Circle Geometry Types he found within his surveys. It is doubtful if anyone would ever have surveyed all of the sites he surveyed had he not done so as a hobby.[MSBB p18]*

## Possible uses of Flattened Circles

Robin Heath has noted an approximate relationship between the perimeter lengths of the type A and B stone circles and the perimeter length of a circle resulting from diameter MN, namely that if the circle were to be taken to equal the solar year of 365.2422 days, then the resulting unit of length of the perimeter of the Type A flattened ring would be 354.367 days, the lunar year, to an accuracy of 99.64% . The Type B flattened ring perimeter similarly has a length corresponding to the eclipse year, 346.62 days, to an accuracy of 99.19%. The A and B flattened circle geometries store these two astronomical constants calibrated with respect to the solar year and each other.

There was probably some other purpose for these geometries related to the limitations and opportunities of a stone age numeracy. For instance the Type A could well have been a geometrical calculator which (probably intuitively) derived the angular unit we call the radian, in which pi is substituted for an angular measuring system like our own degrees, the latter arising by the Sumerian Civilisation, after 3,000 BCE and in the ancient near east.

Thom shows the Type A to be a division of the circle into 30 degree sectors, from the 10 o'clock point M back to the 2 o'clock point F. He also notes that the angle BAC [19.1 degrees] is nearly exactly one third of a radian [*MSB* p28] and this means that the length of the arc on the perimeter (either side of B) equals one third of AB; so if AB = 3 then the arc either side of B is equal to one unit (to one part in 2,500). The diameter AB = 3 would have a circular perimeter equal to 3*pi so that each arc either side of B gives a length which is the diameter for a perimeter equal to AB/3.

What was achieved within the Type A was some sort of recognition that pi was a function translating the diameter into the perimeter. It would have been seen that **two points** on the circumference of a circle form an angle to the circle's **center** that is double the angle seen from a point *opposite* the two points, on the *circumference*. The equilateral triangles, implicit in 5/6ths of the Type A perimeter, divide the circle (and 360 degrees) into segments of 30, 60, 120, and 240 degrees but then the flattened portion divides a new arc into two thirds of a radian, generating arcs related (without pi) to their forming radius AB, i.e. they are effectively commensurate with that radius. In the world of radians, pi is integral to the angular units and thus pi is somewhat removed from the calculation and in this case two arcs were made which are one third of the radius in length and hence subtend angles of 1/3 radian.

It is important to recognise that the likely motivation in moving from fixed geometry to arc lengths was the convenience of measuring angles as a single length rather than as two sides of a right triangle, which form the trigonometrical ratios of sine, cosine and tangent. We measure angles in a similar way when using degrees since any degree scale operates by separating adjacent degree divisions with a constant arc length. Having 360 degrees hides pi from our consideration and by using arc measures, angles could be measured as lengths on reasonably large circular structures which would approach or exceed that of the degree scale and approach or exceed the accuracy of a minute (1/60th) of a degree corresponding with the unaided eye's capabilities. Whilst we make two pi equal to 360 degrees, it appears the Type A builders identified a method for generating a length equal to 1/6*pi the radius.

Sub lengths would be sub angles and these could be stored as measurements using rope or a pre-arithmentic notation.