In my academia.edu paper on lunar simulators, based upon the surviving part of a circular structure at Le Manio (CarnacAn extensive megalithic complex in southern Brittany, western France, predating the British megalithic., Brittany), a very simple but poor approximation to PIor π: The constant ratio of a circle's circumference to its diameter, approximately equal to 3.14159, in ancient times approximated by rational approximations such as 22/7. could be assumed, of 82/26 (3.154) since there seem to have been 82 stones in the circle and the diameter was 26 of the inter-stone distance of 17 inches. The number 82 is significant to simulation of the moon’s orbit since that orbit is very nearly 27 and one third days long (actually 27.32166 days). In three orbits therefore, there are almost exactly 82 days and in day-inch countingThe practice of counting the days, using inches or other small units, between synodic phenomena such as years or planetary loops. that is 82 day-inches. Also of interest is the fact that in three orbits, the exact figure would be 81.965 day-inches which approaches the megalithic rod of 2.5 MY as 6.8 feet.

The value of 82/26 for Pi is not accurate. If one is looking for a circle which has an integer number of 82 inch lengths, a more natural approach is to form a rope of N times 82 inches long and then try to lengthen a radial rope until the circumference fits what the rope generates as a circumference. If N were made equal to six then the target rope length could be formed by adding an 82 inch length to itself five times, to give a length of 410 inches. Dividing this by Pi times two gives a radial rope 65.254 inches which is exactly two of the megalithic yards defined by the three year triangle of Le Manio, to one part in 18,500.

The reason why 82 inches on the circumference is commensurate with megalithic yards on the radius is that the megalithic yardAny unit of length 2.7-2.73 feet long, after Alexander Thom discovered 2.72 ft and 2.722 ft as units within the geometry within the megalithic monuments of Britain and Brittany. is 261/8 inches long because the ratio of 820/216 is very accurately Pi ( to one part in 18,500.)

This value of Pi would naturally emerge if and when circumferences that divide by 82 inches are of interest and since the megalithic yard had been generated but the right-angled triangle of the Quadrilateral (points P-Q-R) as a unit 261/8 or 32.625 day-inches long, being the difference between three lunar years and three solar years, in day-inches.

This has ramifications beyond circular structures since the eggs and flattened circles found by Thom all manifest radial dimensions, related to* *the megalithic yard through the means of their construction in whole numbers of MY within Pythagorean triangles. Once it became clear that integer multiples of 82 inches on the perimeter led to rational radii in megalithic yards, compound radial structures could exploit this numerical characteristic, providing only symmetrical alterations of the circle were involved, as is the case with most eggs and flattened circles.

In the case of the 82-stone circle at Le Manio, the circumference appears to have been chosen to be 17 units of 82 inches giving stones 17 inches apart. This length is then the same as that using 82/13 as an approximation to 2 Pi, namely 1394, whilst the inner radius would then be 221.862 inches or 6.8 megalithic yards, a number that preserves 17 as 17 divided by 2.5.

The megalithic astronomers were nothing if not practical in using the properties of numbers within monuments that were built to demonstrate or measure the time-world of the geocentricThe ancient view of the planetary world in which the inner planets orbit the sun and the sun and planets orbit Earth (the geo part). The Chaldean order was Earth - Moon - (Mercury - Venus -Sun) - Mars - Jupiter - Saturn. sky. By building an 82-stone simulator in which a moon marker moved three stones every day, the astronomers had access to the eclipticThe path of the Sun through the sky along which eclipses of sun and moon can occur, traditionally divided into the 365¼ parts of the solar year, each part then a DAY in angle rather than time. model of the ZodiacThe 12 constellations through which the sun passes in the solar year of 365.2422 days within which the Moon sat on successive evenings. In India, dates came to be recorded as the moon in a given natshatra of 27 bright star constellations perhaps echoing the 27 days in the orbit.

With such a simulator it became possible to observe the movement of the lunar orbital nodes over 18.618 years (6800 days), at right angles from where the moon was most above or most below the ecliptic path of the sun, or when a lunar eclipse occurred at the node. These capabilities could be combined to lunar alignmentsIn general, to the sun and moon on the horizon, rising in the east or setting in the west.

Also, a name special to Carnac's groups of parallel rows of stones, called Le Menec, Kermario, Kerlescan, and Erdevan. between maxima and minima (standstills) and form the basis for calendrical counting, simulation and prediction.

[1] Equal to the 1394 inch circumference of the 82 stone simulator at Le Manio, if the “megalithic rod” used were 82 inches long rather than being 2.5 megalithic yards of 32.625 inches equalling 81.56 inches, half an inch the lesser.