With regard to the circle that appeared at Hackpen Hill, nr Broad Hinton, Wiltshire (reported on 31st of July), I have the following Observations to offer:

This seems to be yet a further example providing a simple means of Squaring the Circle. As the accompanying graphic will show, the large central circle exactly fits into the overall diameter 5 times. More important however, the smaller circle (inscribed four times around the circumference) exactly fits the overall diameter 7 times. (The proportion of the larger to the smaller is therefore 7/5). If we regard the overall radius of the formation to equal 1, then the overall circumference will be equal to 2*pi.

If the overall radius (as displayed by the upper vertical axis) is now divided into 7 smaller circles (shown here in orange), each of these 7 smaller circles = 1/7 of the radius. If the value of the radius itself is counted as being equal to 1, then each of these small circles will have a value of (1/7)*1. Moving upwards from the centre, if small circle number 6 is bisected (i.e. subdivided into two smaller circles), then the distance from the centre of the formation to this point of bisection will equal 5.5/7 of the overall radius.

Now if a square is drawn where the top face cuts this point of bisection, we find that half of one side of the square will be equal to 5.5 of the seven small-circle diameters. This means that two whole sides of the square will now be equal to 22/7, and therefore all four sides will be equal to 2*(22/7), thereby equalling 2*pi, being also the value of the radius of the large circle.

It is not, of course, completely accurate (3.1429 as opposed to 3.14159), but is the nearest achievable through simple geometry.

Roger Wibberley