*December 2016 in numbersciences.org Hits: 3872*

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.

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 metrologyThe application of units of length to problems of measurement, design, comparison or calculation.), [identified first by John Michell in *Ancient Metrology* and then John Neal in *All Done With Mirrors*] include **176/175Ratio crucial to maintaining integers (see geometry lesson 2) between radii and circumference of a circle, and crucial to the micro-variation of foot modules in ancient metrology.**. 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 22/21. 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 It is uncontested (though not peer reviewed) that Le Manio Quadrilateral used inches to count days to resolve the megalithic yardA unit of length around 2.7 to 2.73 feet long, as the day-inch difference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 footThe standard prehistoric foot (of 12 inches) representing a unity from which all other foot measures came to be formed, as rational fractions of the foot, a fact hidden within 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.

Stop Press: Since this article was written, the archaeologists have indeed proposed that chalk drums were employed to easily lay out lengths at Stonehenge. Please see the Times article.

I was fortunate to recover this article from the Wayback Machine after much searching since it was destroyed when the RAID backup failed.