Three Folkton Chalk Drums found in a young girl's grave {PHOTO: TRUSTEES OF THE BRITISH MUSEUM]

Perhaps as early as 4000 BC, there was a tradition of making chalk drums. Three highly decorated examples were found in a grave dated between 2600 and 2000 BC in Folkton, northern England and one undecorated chalk drum in southern England at Lavant in an upland downs known for a henge and many other neolithic features discovered in a recent community LIDAR project. The Lavant LIDAR project and the chalk drum found there are the first two articles in PAST, the Newsletter of The Prehistoric Society. (number 83. Summer 2016.) It gives the height and radius of both the Folkton drums 15, 16 and 17 and the Lavant drum, presenting these as a graph as below.

Adapted graphic showing inches as well as mm, and possible PI relationships for the chalk drum diameters,
key to the fact that such drums can be rolled.In line with megalithic numeracy,
the simple yet accurate value of 22/7 for PI is shown.

Local chalk is a relatively easily carved yet substantial material and a cylindrical drum can be rolled and, being given a definite diameter, causes the circumference to travel a known distance on the earth. Folkton 17 has a 4 inch diameter which gives an 88/7 inch circumference (using PI = 22/7) which equals 22/21 feet of twelve inches. In Sacred Number and the Lords of Time, 167-171, I point out that the microvariations to be found between measures of the same module (in historical metrology), [identified first by John Michell in Ancient Metrology and then John Neal in All Done With Mirrors] include 176/175. This ratio is the product of two early versions of PI since 176/175 = 8/25 times 22/7. leading to the fact that 1/3 foot diameter (= 4") will give a circumference of 22/21 feet. (see panel). The module is 25/24 feet, varied by 176/175 to give 176/175. Since the accurate PI of 22/7 is present in the ratio 176/175, then a circumference of 22/21 feet times 7/22 gives a diameter of 22/21 feet times 7/22 equalling a 1/3 foot diameter or the 4 inch diameter of Folkton 17.

There are three levels of interpretation:

• Firstly, the four inch radius appears to imply that inches were the native units of measure and the five inch radius of Folkton 16 appears to support this. It is uncontested (though not peer reviewed) that Le Manio Quadrilateral used inches to count days to resolve the megalithic yard, as the day-inch difference when counting three lunar and three solar years.
• Secondly, if the later megalith builders had evolved a network of fractional measures based upon the English foot as unit, 1/1, then new measures were made from that foot by laying out right triangles in feet, with a different number of whole units in the two longest sides, numbers we would call the numerator and the denominator of a fraction. In the case of 22 feet for hypotenuse and 21 for base, the 21 divisions of the base can be made to rise at right angles to define 21 divisions on the hypotenuse, each 22/21 feet long. The chalk drum would perform 21 revolutions in travelling the 22 feet of such a hypotenuse. The diameter has to be 1/3 of a foot so that 1/3 times 22/7 equals 22/21 feet.
• Thirdly, if one wishes to make a circumference of 22 feet or 21 rotations of this chalk drum, then the diameter must be 21 times 1/3 feet or seven feet. 22/21 is in fact the Thoth ratio found between the 1/6 arc on the circumference relative to the straight distance between the ends of the arc. Egyptian Thoth presents this in his iconography, because PI was a sacred invariant to geometers and this brings us to decorated drums being found in a high status burial, if the family involved were the geometers who laid out megalithic monuments and pathways. The undecorated drum found at Lavant shows signs of usage as if it had been rolled many times.

The largest of the three drums, Folkton 15, appears to have a circumference of 18 inches and, if so, the drum transfers the idea of rationality to the circumference so that the diameter is an irrational number of inches, 7 and 9/11th inches. Such a drum would be able to lay out cubits of 3/2 = 1.5 feet in a line, enabling yards to work extensively. One also notices that the designs on the tops of drums has a possible role in dividing the rotation of the drum like a rotary ruler or in the angular sense.

Picture of Lavant Chalk Drum, showing a plain drum
with a single boss and double rim on its side separated by a depression.

grabbed with thanks from 3D SketchFab by Marta Diaz-Guardamino

The Lavant drum, shown above, appears to have been a practical rolling device equivalent to a short and fat axle with no wheels.

1. The radius is measured to be 115mm or 4.528 inches (in our notation).
2. Multiplied by 22/7 this is 14.23 inches or 1.18578 feet.

To understand this length requires that the largest type of foot using in ancient metrology, called the Russian foot but essentially 7/6 (1.16feet which, after two yards, becomes seven English feet. But here the geographical variant is used of 7/6 times 1.01376, the Geographical Constant.

Geographical constant is a term within the modern subject called Ancient Metrology [see Michell. 1981. and Neal. 2000.], so-called because it defines an important relationship between the equator of the earth and its mean circumference. In brief,

1. the sun moves one day in angle every day and that angular distance can be measured on the equator (from other parallels of latitude) as being 360,000 feet.
2. the mean radius of the earth can be known from the polar radius and equatorial radius and each degree of latitude on the mean earth would then be 69.12 miles long.

Despite the north-south deformation of the earth (due to its daily rotation) one degree of latitude, between 51 and 52 degrees, corresponds to the mean earth in measuring 69.12 miles. Within this latitude lies Stonehenge and Avebury, separated by one quarter of this degree length. Lavant lies just below 51 degrees. So why be using the geographical constant at Lavant?

John Michell, in The Dimensions of Paradise and elsewhere, deconstructed the geographical constant as being 3168/3125 and observed that temples in prehistory were metrologically organised to express 3168 units around their perimeter, making such structures models of the mean earth which, by inference, gives signifies the mean earth as being the spiritual earth. Not only were monuments being built within or near the degree of latitude corresponding to the mean degree within southern Britain, but also those monuments needed to be laid out so that their circumferences could correspond to the 3168 units required, and using the geographical units of measure corresponding to that mean degree. In turn, to understand the Lavant chalk drum we might need to deconstruct 3168 as being 22 times 144.

The number 3168, in being 144 x 22, is to be the perimeter of sacred spaces which in circular monuments means the circumference needs to be 22 times 144 units long. The diameter of the monuments, calculated as 3168 divided by 22 and multiplied by 7 must give a required radius of 72 (144/2) times 7 which equals 1008. (Using 2 PI = 44/7)

Since six revolutions of the chalk drum gave seven geographical feet, 72 x 6 rotations would have defined a rope from the central peg which could achieve a perimeter of 3168 geographical feet when used to define a circle. The foot of 7/6 is revealed to be ideal in solving a particular problem of generating on the ground a radius which a pegged rope can turn into a perimeter iof 3168, representing a part of the actual earth as the mean earth, seen to be sacred as part of the mean earth.

### Conclusions

The chalk drums are probably evidence of their use  in southern Britain as metrological contraptions whose power lay in their natural expression of PI as 22/7.

1. Folkton 17 (4" diameter) allowed the creation of a unique variation of 22/21 feet on its circumference when rolled. This ratio is one third of PI and
1. this allowed one third of a foot to transform 22/7 into 22/21
2. a foot of 22/21 enabled a hexagon to be constructed with sides 21 feet within a circle whose circumference would then be
2. Folkton 16 (5" diameter) is 5/4 = 1.2 of these feet in circumference and hence could be useful within the same module
3. Folkton 15 appears designed to provide an explicit cubit of 1.5 feet (18") with every rotation of travel by being (7 x 18)/22 (= 63/11) inches in diameter.
4. The undecorated and worn Lavant drum seems contrived to generate 7/6 geographical feet in circumference so as to allow the formation of circular perimeters for sacred spaces measuring 3168 geographical feet or multiple thereof, an activity connected to its latitude and perimeters of other contemporary structures.

The drums appear to be within a range of sizes large enough to be accurate and small enough to be simply used in the task of laying out distances on the earth, perhaps by a single trained individual who belonged to a craft tradition and this may have led to a premature death being marked in the burial of three decorated chauk drums at Folkton, up to a thousand years after the Lavant drum was disposed of. If so then the drums may represent the iconography of the times perhaps representing PI or a god connected to geodetic constructions. Further examples could strengthen this interpretation.

An interactive demonstration of how accurate rational values of PI can be formed using denominators one to ten.
The last three all appear in ancient metrology; 22/7 (the best), 25/8 and 2 PI of 63/10.
[using oCanvas with thanks to Johannes Koggdal]

There are many good rational versions of PI but 22/7 is remarkably good and arrives early in the number field, making it amenable to ancient forms of numeracy.